ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.10
https://doi.org/10.26493/1855-3974.2730.6ac
(Also available at http://amc-journal.eu)
Braid representatives minimizing the number of
simple walks*
Hans U. Boden † , Matthew Shimoda ‡
Mathematics & Statistics, McMaster University, Hamilton, Ontario, Canada
Received 15 November 2021, accepted 19 July 2022, published online 21 November 2022
Abstract
Given a knot, we develop methods for finding a braid representative that minimizes
the number of simple walks. Such braids lead to an efficient method for computing the
colored Jones polynomial of the knot, following an approach developed by Armond and
implemented by Hajij and Levitt. We use this method to compute the colored Jones poly-
nomial in closed form for the knots 52, 61, and 72. The set of simple walks can change
under reflection, rotation, and cyclic permutation of the braid, and we prove an invariance
property which relates the simple walks of a braid to those of its reflection under cyclic
permutation. We study the growth rate of the number of simple walks for families of torus
knots. Finally, we present a table of braid words that minimize the number of simple walks
for knots up to 13 crossings.
Keywords: Knots, braids, simple walk, colored Jones polynomial.
Math. Subj. Class. (2020): 57K10, 57K14
1 Introduction
The Jones polynomial VLptq is an invariant of knots and links defined using quantum rep-
resentations of braids. It can be uniquely characterized as the polynomial-valued invariant
of oriented links with V⃝ptq “ 1 for ⃝ the unknot and satisfying the skein relation
t´1VL` ptq ´ tVL´ ptq “ pt1{2 ´ t´1{2qVL0ptq,
*The authors are especially grateful to Alexander Stoimenow for providing crucial input. They would also like
to thank Homayun Karimi, Robert Osburn, Andrew Nicas, Will Rushworth, and Cornelia Van Cott for valuable
feedback.
†Corresponding author. The author would like to acknowledge funding from the Natural Sciences and Engi-
neering Research Council of Canada.
‡The author acknowledges funding from a USRA award and a Stewart award from McMaster University.
E-mail addresses: boden@mcmaster.ca (Hans U. Boden), mattshimoda@hotmail.com (Matthew Shimoda)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 23 (2023) #P1.10
where L`, L´, L0 are identical outside a neighborhood, where they are as pictured
L` L´ L0.
The Jones polynomial admits a combinatorial state sum formula that can be used to com-
pute it, but the complexity of the computation grows exponentially with the crossing num-
ber. So this method is impractical for computations that involve links with a large number
of crossings.
The colored Jones polynomial is a powerful knot invariant packaged as a sequence of
Laurent polynomials JN,Kpqq for N ě 2, with N “ 2 giving the usual Jones polynomial.
It encodes subtle geometric information about the knot complement and appears in several
famous open problems in quantum topology, including (i) the Volume Conjecture [18, 23];
(ii) the Slope Conjecture [12, 17]; and (iii) the AJ Conjecture [11]. The first relates the limit
of JN,Kpqq for q “ e2πi{N as N Ñ 8 to the hyperbolic volume of the knot complement;
the second posits that every Jones slope of a knot is the slope of an incompressible surface
in the knot complement; and the third asserts that the recurrence relation for the N -th
colored Jones polynomials is given by the A-polynomial of [9], a plane curve associated to
the character variety of SLp2,Cq representations of the knot group.
For further background information on the colored Jones polynomial and its relation to
the geometry of 3-manifolds, we refer the reader to the books [24] and [10] and their ex-
tensive bibliographies. The colored Jones polynomial also has intriguing number theoretic
interpretations that will not be discussed in this paper; for more details about these aspects,
we refer the reader to the recent papers [3, 4, 20, 21] and their bibliographies.
Our goal in this paper is to study a probabilistic method for computing the colored
Jones polynomial. This approach was first developed by Huynh and Lê in [15], and it
was later described in terms of walks along braids by Armond [2]. Armond identified
the special role played by simple walks, resulting in an extremely efficient algorithm for
computing JN,Kpqq, which has been implemented by Hajij and Levitt [14]. The algorithm
is exponential in the number of simple walks on the braid, so it is natural to try minimize
the number of simple walks before executing the program of [14]. However, as we shall
see, this number is highly dependent on the braid representative chosen.
We study how the number of simple walks changes under taking reflection, rotation,
and cyclic permutation of a given braid. We also examine the growth rate of the number
of simple walks for two families of torus knots. For instance, for the family of p2, nq torus
knots, the simple walks satisfy a Fibonacci recurrence and grow exponentially in n. For
the family of p3, nq torus knots, the simple walks satisfy a tribonacci recurrence and also
grow exponentially in n. We further prove that the total number of simple walks on a braid
and its reflection is invariant under cyclic permutation. This fact is used to facilitate finding
braid representatives with the least number of simple walks. For knots up to 13 crossings,
we developed a program that finds minimal braid representatives. When these braids are
used in conjunction with the program of Hajij and Levitt [14], this provides an efficient
method for computing the colored Jones polynomial for these knots.
We close this section with a brief synopsis of the rest of this paper. In Section 2, we re-
view the method from [2, 15] for computing the colored Jones polynomial. In Section 3, we
use it to compute JN,Kpqq in closed form for the knots 52, 61, and 72. These computations
were originally performed by Masbaum using skein theory, see [22]. In Section 4, we recall
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 3
the basic results about braid representatives for knots and study the effect of the Markov
moves on the set of simple walks. In Section 5, we introduce the set of semi-simple walks,
and we show that it is invariant under cyclic permutation of the braid word. In Sections 6
and 7, we study the growth rate of the number of simple walks for two families of torus
knots. In Section 8, we present the output of a program for finding braid representatives
that minimize the number of simple walks.
2 The colored Jones polynomial and walks along braids
Given a knot K and integer N ě 2, the colored Jones polynomial JN,Kpqq is a Laurent
polynomial in the variable q1{2. It is normalized so that JN,⃝pqq “ 1, where ⃝ is the un-
knot. When N “ 2, the colored Jones polynomial agrees with the usual Jones polynomial.
In general, the N -th colored Jones polynomial of a knot K can be expressed in terms of
the usual Jones polynomial of the pN ´ 1q strand cable of K. However, since the crossing
number of the pN ´ 1q strand cable of a knot is pN ´ 1q2 times the crossing number of the
knot, this does not lead to a practical method for computing the colored Jones polynomial.
One approach for computing the colored Jones polynomial is presented by Huynh and
Lê [15]. Starting with a braid β whose closure is the given knot, Huynh and Lê use meth-
ods from quantum algebra to express the colored Jones polynomial as the inverse of the
quantum determinant of an almost quantum matrix. The matrix is constructed through the
product of Burau matrices, which we obtain from the crossing and orientation properties of
β.
A second approach is presented by Armond [2]. It is based on a probabilistic inter-
pretation of the colored Jones polynomial and involves counting walks along braids. This
method is closely related to the previous one, and in fact it provides a visual representation
of the quantum algebra approach. The idea is to view walks along the braid as traversing
the strands of the braid from the bottom to the top and to record information about the
crossings and their orientations as a product of operators. The end result is the same as
that obtained by taking the quantum determinant of the deformation of Burau matrices, but
Armond’s approach is more accessible and requires less background material on operator
theory. One interesting aspect is that the complexity of the computation is sensitive to the
choice of braid word, and this will be explored further in Section 4. For now, we focus on
describing Armond’s approach and the special role played by the simple walks.
We begin by introducing a little terminology from braid theory.
Definition 2.1. A braid is a set of m strands running from top to bottom with no reversals
in vertical direction. The strands may cross each other, but only two strands can participate
at each crossing.
Given a braid β, a braid word is an expression of the form
β “ σε1i1 σ
ε2
i2
. . . σεℓiℓ ,
where εi “ ˘1 and σi is a symbol. Braid words are read from left to right, and braids are
drawn from top to bottom. For 1 ď i ď m ´ 1, σi represents the braid with one crossing
where the pi`1q-st strand crosses over the i-th strand. The inverse σ´1i represents the braid
where the i-th strand crosses over the pi ` 1q-st strand.
4 Ars Math. Contemp. 23 (2023) #P1.10
The braid word β “ σε1i1 σ
ε2
i2
. . . σεℓiℓ has ℓ crossings, and we say it has braid length ℓ.
If β is a braid on m strands, we say it has braid width m. Note that braid words are not
uniquely determined by the braid. Applying a braid relation (see below) will alter the word
without changing the braid. The writhe of a braid is defined to be the sum of the signs on
all its crossings. For example, the braid word above has writhe wpβq “
řℓ
i“1 εi.
The braid group on m strands is denoted Bm. Abstractly, it is the group with generators
σ1, . . . , σm´1 and relations
(i) σiσj “ σjσi for 1 ď i, j ď m ´ 1 with |i ´ j| ą 1 and
(ii) σiσi`1σi “ σi`1σiσi`1 for 1 ď i ď m ´ 2.
Relation (i) is called far commutativity and (ii) is called the Yang-Baxter relation (or braid
relation). The group operation is given by concatenation of words or, equivalently, by
stacking geometric braids, one on top of the other.
Next, we introduce the notions of paths and walks along braids. A path starts at the bot-
tom of the braid and traverses arcs of the braid, sometimes jumping down, until it reaches
the top of the braid. If the path starts at strand i on the bottom and ends at strand j at the
top, we say it is a path from i to j. Whenever the path encounters a crossing, if it is on the
overstrand, it is allowed to jump down to the undercrossing arc. If it is on the understrand,
then it must stay on that strand. At each crossing the path encounters, a weight from the
set tai,εi , bi,εi , ci,εi | i “ 1, . . . , ℓu is assigned. The weight will depend on the crossing,
its sign, and the arcs traversed by the path at that crossing. For example, if the path jumps
down at the i-th crossing, it is assigned the weight ai,εi . Otherwise, it is assigned the weight
bi,εi if the path traverses the understrand and ci,εi if it traverses the overstrand. Note that,
at a given crossing, the path will follow the braid unless it jumps down there. The total
weight of the path is the product of the weights of the crossings.
A walk W along β consists of a set J Ď t1, . . . ,mu, a permutation π of J , and a
collection of paths with exactly one path in the collection from j to πpjq for each j P J .
The weight of a walk is p´1qp´qq|J|`invpπq times the product of the weights of the paths,
where |J | is the cardinality of J and invpπq is the number of inversions in π, i.e., the
number of pairs of elements in J with i ă j and πpiq ą πpjq.
The paths in a walk are ordered from left to right, using their starting strand at the
bottom of the braid. This induces an ordering on the weights. The order of the weights is
important, and that is because the operators associated to the weights are non-commuting.
Operators based at different crossings do commute, but operators at the same crossing do
not. Thus, the effect of non-commutativity can be computed locally at each crossing of
the braid. Given a walk W and a crossing i of the braid, we use Wpiq to denote the local
weight of paths of W through crossing i in the given order. If no path of W passes through
crossing i, then we set the local weight Wpiq “ 1.
A stack of walks is any ordered collection W1 ¨ ¨ ¨Wk of walks. Visually, this can be
viewed as stacking the walks on top of one another with W1 at the top and Wk on the
bottom. The weight of the stack is the product of the weights of the paths in the appropriate
order. If two paths belong to different walks, then the path in the higher walk is multiplied
to the left of the path in the lower walk. If two paths belong to the same walk, then the path
which begins to the left of the other path is said to be above and is multiplied to the left
of the other path. Given a stack S “ W1 ¨ ¨ ¨Wk and a crossing i of the braid, we let Spiq
be the local weight of the stack at i; it is equal to the product pW1qpiq ¨ ¨ ¨ pWkqpiq of local
weights of the walks of the stack at i in the given order.
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 5
With walks along braids established, we now show how they can be used to com-
pute the colored Jones polynomial. Let R “ Zrq, q´1s. Let x̂ and τx be operators on
the ring Rrx˘1, y˘1, u˘1s given by x̂fpx, y, . . . q “ xfpx, y, . . . q and τxfpx, y, . . . q “
fpqx, y, . . . q. The operators ŷ, û, τy , τu are defined similarly. We associate operators to
each of the crossing weights using the formulas:
a` “ pû ´ ŷτx´1qτy´1, b` “ û2, c` “ x̂τy´2τu´1,
a´ “ pτy ´ x̂´1qτx´1τu, b´ “ û2, c´ “ ŷ´1τy´1τu.
By taking the summation of all walks on the braid and writing their weights in terms of
the above operators, we obtain the operator P . Letting P act on the constant 1 and making
the substitutions x “ z, y “ z and u “ 1, we obtain a polynomial EpP q. Let EN pP q
denote the polynomial obtained by making the substitution z “ qN´1 to EpP q.
The next result shows how to compute the colored Jones polynomial of a knot from its
braid representative. It was proved by Huynh and Lê in [15] and appears as Theorem 2.3
in [2].
Theorem 2.2. Let K be a knot obtained as the closure of a braid β P Bm. Its colored
Jones polynomial is given by
JN,Kpqq “ qpN´1qpwpβq´m`1q{2
8
ÿ
n“0
EN pPnq, (1)
where the operator P is the sum of the weights of the walks on β with J Ď t2, . . . ,mu.
Stacks of walks are produced when we take the product of the weights of walks. This
occurs in the step when we expand the operator P to the power of n.
Huynh and Lê also gave a useful method for evaluating the terms EN pPnq which avoids
operator theory. The key result is the following lemma from [15] which computes EN pPnq
directly from the weights once they have been put in a preferred order. In the following,
we suppress the dependence of the weights on the crossing and write a˘, b˘, c˘ instead of
ai,˘, bi,˘, ci,˘.
Lemma 2.3.
EN pbs`cr`ad`q “ qrpN´1´dq
d´1
ź
i“0
p1 ´ qN´1´r´iq
EN pbs´cr´ad´q “ q´rpN´1q
d´1
ź
i“0
p1 ´ qr`i`1´N q
We will apply this lemma to the local weights Spiq of each stack at each crossing.
However, before we can apply Lemma 2.3, we must first put the local weights at a crossing
into the preferred order. This can always be achieved using the following relations:
a`b` “ b`a`, a`c` “ qc`a`, b`c` “ q2c`b`
a´b´ “ q2b´a´, a´c´ “ q´1c´a´, b´c´ “ q´2c´b´
6 Ars Math. Contemp. 23 (2023) #P1.10
Once the local weights have been put into the preferred order at each crossing, The-
orem 2.2 and Lemma 2.3 can be applied locally at each crossing to compute the colored
Jones polynomial.
The computation is simplified by the observation that only simple walks contribute to
the colored Jones polynomial [2, Lemma 2.5(a)]. Here, a walk is said to be simple if no two
paths intersect in the walk, otherwise it is non-simple. It turns out that non-simple walks
occur in cancelling pairs, so for the purpose of computing JN,Kpqq, it is enough to consider
only simple walks.
The computation is further simplified by the fact that, for any stack of walks, the evalu-
ation of its weights will vanish if the walks in the stack traverse the same arc on N or more
different levels [2, Lemma 2.5(b)]. This is extremely useful because it reduces the com-
plexity of the computation and guarantees that only finitely many terms of
ř8
n“0 EN pPnq
contribute to the N -th colored Jones polynomial. In particular, for a fixed N , there will
always be an upper bound to the integers n which need to be considered in the infinite sum
of Equation (1). In practice, this bound can be determined by comparing the arcs traversed
by the set of all simple walks for a given braid word.
3 The colored Jones polynomial in closed form
In this section, we apply Theorem 2.2 and Lemma 2.3 to compute JN,Kpqq, the colored
Jones polynomial, for the knots 52, 61 and 72. This is achieved by choosing favorable braid
representatives, namely those with only a few simple walks.
In [22], Masbaum uses skein theory to compute the colored Jones polynomial for all
twist knots, a class which includes 52, 61, and 72. More general calculations of the colored
Jones polynomial for the double twist knots can be found in [20, 21].
  
Figure 1: The simple walks A and B shown as arcs with zebra stripes for the braid
σ´12 σ1σ
3
2σ1 with closure the knot 52.
Example 3.1. The braid word σ´12 σ1σ32σ1 represents the knot 52 and has two simple walks
with J Ď t2, 3u. They are A “ qa1,´c3,`a4,`b5,` and B “ q3b1,´c1,´c2,`c3,`a4,`
b5,`b6,` (see Figure 1). Notice that A has J “ t3u and B has J “ t2, 3u. Since the walks
A and B both traverse the third strand at top and bottom, stacks only need to be considered
up to level N ´ 1.
Using Theorem 2.2, we can write the colored Jones polynomial as the following:
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 7
JN,Kpqq “ qp1´Nq
N´1
ÿ
n“0
EN ppA ` Bqnq,
“ qp1´Nq
N´1
ÿ
n“0
EN ppqa1,´c3,`a4,`b5,` ` q3b1,´c1,´c2,`c3,`a4,`b5,`b6,`qnq.
We will expand the above expression using the q-binomial theorem. For that purpose,
we introduce the Gaussian binomial coefficients (or q-binomial coefficients), which are
defined by
ˆ
n
k
˙
q
“
k´1
ź
i“0
ˆ
1 ´ qn´i
1 ´ qi`1
˙
.
The expansion of the above expression includes a sum of products of A’s and B’s, which
can be interpreted as stacks.
To apply Lemma 2.3, the weights at each crossing must be placed into the order bscrad,
and it is preferable to expand pA`Bqn as a sum of terms of the form BkAn´k. Since the lo-
cal weights at different crossings commute, the only potential issue with non-commutativity
of A and B is at the first crossing. Since a1,´b1,´c1,´ “ qb1,´c1,´a1,´, we have AB “
qBA, so we can adjust for inversions using the q-binomial coefficient:
pA ` Bqn “
n
ÿ
k“0
ˆ
n
k
˙
q
BkAn´k.
Next we use Lemma 2.3 to apply EN p¨q to evaluate each stack. First, these walks
have weights bi,` indexed alone at the fifth and sixth crossings, which evaluates to 1.
Additionally, the ci,˘ weights at the first and second crossings in walk B always cancel
out since EN pc1,´q “ q´pN´1q and EN pc2,`q “ qN´1. We still have c3,` in each walk A
and walk B. Therefore, for each walk in a stack, the term A or B, qN´1 is contributed.
Similarly, the weight a4,` appears in both walks, so a stack consisting of n walks will
contribute
śn
i“1p1 ´ qN´iq. Meanwhile, the weight a1,´ is only in walk A, so the stack
BkAn´k will contribute
śn´k
i“1 p1 ´ qn`i´N q. Additionally, we need to adjust for the
correct order of b1,´ and c1,´ in the weights in products containing Bk. The number of
times the relation is applied increases quadratically with the exponent of B. That is, for a
weight containing Bk, its contribution is qk
2´k. Finally, the remaining powers of q arise
from the existing variables in the simple walks.
Applying Theorem 2.2, the colored Jones polynomial of 52 can be written in closed
form as
JN,Kpqq “ qN´1
N´1
ÿ
n“0
n
ÿ
k“0
ˆ
n
k
˙
q
qnN`kpk`1q
n
ź
i“1
`
1 ´ qN´i
˘
n´k
ź
i“1
`
1 ´ qn`i´N
˘
.
Example 3.2. The braid word σ1σ2σ1´1σ3´1σ2σ3´1σ1 represents the knot 61 and has
three simple walks with J Ď t2, 3, 4u. They are A “ qa4,´a6,´, B “ q3c2,`a3,´a4,´b5,`
b6,´c6,´ and C “ q5c1,`c2,`b3,´c3,´a4,´b5,`b6,´c6,´b7,` (see Figure 2). Notice that A
has J “ t4u, B has J “ t3, 4u, and C has J “ t2, 3, 4u. Since the walks A,B and C all
traverse the fourth strand at top and bottom, stacks only need to be considered up to level
N ´ 1.
8 Ars Math. Contemp. 23 (2023) #P1.10
   
Figure 2: The simple walks A,B and C shown as arcs with zebra stripes for the braid
σ1σ2σ1
´1σ3
´1σ2σ3
´1σ1 with closure the knot 61.
Using Theorem 2.2, the colored Jones polynomial for 61 can be written as
JN,Kpqq “ qp1´Nq
N´1
ÿ
n“0
EN ppA ` B ` Cqnq ,
“ qp1´Nq
N´1
ÿ
n“0
EN
`
pqa4,´a6,´ ` q3c2,`a3,´a4,´b5,`b6,´c6,´
` q5c1,`c2,`b3,´c3,´a4,´b5,`b6,´c6,´b7,`qn
˘
.
We use the q-multinomial theorem to expand the terms pA ` B ` Cqn, and to that end
we recall the definition of the q-multinomial coefficients.
   
Figure 3: The simple walks A,B and C as arcs with zebra stripes on the braid
σ1σ2σ
3
3σ
´1
1 σ2σ1σ
´1
3 with closure the knot 72.
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 9
Given an integer r ě 1 and sequence of nonnegative integers m1,m2, . . . ,mr such
that n “ m1 ` ¨ ¨ ¨ ` mr, let
ˆ
n
m1,m2, . . . ,mr
˙
q
“ rnsq!rm1sq!rm2sq! . . . rmrsq!
, (2)
where rnsq “ qn´1 ` ¨ ¨ ¨ ` q ` 1 and rnsq! “ r1sq ¨ ¨ ¨ rnsq . The term
`
n
m1,m2,...,mr
˘
q
is
called the Gaussian multinomial coefficient or q-multinomial coefficient.
With the weights indexed at the third and sixth crossings, the natural order of the walks
is C,B,A, as this will allow use of Lemma 2.3 with only a few other adjustments. The use
of the q-multinomial coefficient is suitable for this order because inversions with respect to
the alphabet C,B,A are reversed with multiplication by q. That is, AB “ qBA, AC “
qCA and BC “ qCB. The expansion of the trinomial takes the form
pA ` B ` Cqn “
n
ÿ
m“0
m
ÿ
k“0
ˆ
n
n ´ m,m ´ k, k
˙
q
Cn´mBm´kAk, (3)
where
`
n
n´m,m´k,k
˘
q
denotes the q-trinomial coefficient of Equation (2).
We can now expand the trinomial for any power n and use Lemma 2.3 to apply EN p¨q
and evaluate each stack. The b˘ and c˘ weights are evaluated the same as in Example 3.1.
The weight a4,´ appears in each of A,B,C, so for every term Cn´mBm´kAk, it will
contribute
śn
i“1p1 ´ qi´N q. Meanwhile, the weight a3,´ only appears in the walk B,
so for the term Cn´mBm´kAk, it contributes
śm´k
i“1 p1 ´ qn´m`i´N q. Similarly, the
weight a6,´ only appears in the walk A, so for the term Cn´mBm´kAk, it contributes
śm´k
i“1 p1 ´ qn´k`i´N q.
Finally, we need to adjust for the correct order of the terms b6,´ and c6,´ in the products
containing Cn´mBm´k. The number of times the relation is applied increases quadrati-
cally with the sum of the exponents of B and C, which is pn´mq`pm´kq “ n´k. That
is, for the term Cn´mBm´kAk, applying the relation introduces a factor of qpn´kq
2´pn´kq.
We follow the same logic for products Cn´m to adjust for the order of the weights b3,´
and c3,´ at the third crossing.
Applying Theorem 2.2, the colored Jones polynomial of 61 can be written in closed
form as
JK,N pqq “ q1´N
N´1
ÿ
n“0
n
ÿ
m“0
m
ÿ
k“0
ˆ
n
n ´ m,m ´ k, k
˙
q
q3n´k´m`pn´kq
2`pn´mq2
ˆ
n
ź
i“1
p1 ´ qi´N q
m´k
ź
i“1
p1 ´ qn´m`i´N q
k
ź
i“1
p1 ´ qn´k`i´N q.
Example 3.3. The braid word σ1σ2σ33σ
´1
1 σ2σ1σ
´1
3 represents the knot 72 and has three
simple walks with J Ď t2, 3, 4u. They are A “ qc3,`a4,`b5,`a9,´, which has has
J “ t4u; B “ q3c2,`c3,´a4,`b5,`a6,´b7,`b9,´c9,´, which has has J “ t3, 4u; and
C “ q5c2,`c3,`a4,`b5,`b6,´c6,´b7,`b8,´b9,´c9,´, which has J “ t2, 3, 4u (see Fig-
ure 3). Since the walks A,B and C all traverse the fourth strand at top and bottom, stacks
only need to be considered up to N ´ 1.
10 Ars Math. Contemp. 23 (2023) #P1.10
We use Theorem 2.2 to write the colored Jones polynomial for 72 as:
JN,Kpqq “ qp1´Nq
N´1
ÿ
n“0
EN ppA ` B ` Cqnq ,
“ qp1´Nq
N´1
ÿ
n“0
EN
`
pqc3,`a4,`b5,`a9,´ ` q3c2,`c3,´a4,`b5,`a6,´b7,`b9,´c9,´
` q5c2,`c3,`a4,`b5,`b6,´c6,´b7,`b8,´b9,´c9,´qn
˘
.
With the weights indexed at the sixth and ninth crossings, the most natural order of the
walks is C,B,A. Note that AB “ qBA, AC “ qCA and BC “ qCB. Therefore, the
expansion of the trinomial is as given in equation (3) above.
We expand the trinomial for all powers of n and use Lemma 2.3 to apply EN p¨q and
evaluate each stack. The b˘ and c˘ weights are evaluated the same as in Examples 3.1
and 3.2. The weight a4,` appears in each of A,B,C, so for every term Cn´mBm´kAk,
it will contribute
śn
i“1p1 ´ qN´iq. Meanwhile, the weight a6,´ only appears in the walk
B, so for the term Cn´mBm´kAk, it contributes
śm´k
i“1 p1 ´ qn´m`i´N q. Similarly, the
weight a9,´ only appears in the walk A, so for the term Cn´mBm´kAk, it contributes
śk
i“1p1 ´ qn´k`i´N q.
Finally, we need to adjust for the correct order of the terms b9,´ and c9,´ in the products
containing Cn´mBm´k. The number of times the relation is applied increases quadrati-
cally with pn ´ mq ` pm ´ kq “ n ´ k. That is, for the term Cn´mBm´kAk, applying
the relation introduces a factor of qpn´kq
2´pn´kq. We follow the same logic for products
Cn´m to adjust for the weights at the sixth crossing.
Applying Theorem 2.2, the colored Jones polynomial of 72 can be written in closed
form as
JK,N pqq “ qN´1
N´1
ÿ
n“0
n
ÿ
m“0
m
ÿ
k“0
ˆ
n
n ´ m,m ´ k, k
˙
q
qnN`2n´m´k`pn´kq
2`pn´mq2
ˆ
n
ź
i“1
p1 ´ qN´iq
m´k
ź
i“1
p1 ´ qn´m`i´N q
k
ź
i“1
p1 ´ qn´k`i´N q.
For hyperbolic knots, the Volume Conjecture asserts that
lim
NÑ8
log |JN,Kpe2πi{N q|
N
“ VolpS
3 ∖Kq
2π
.
For more information about this important open problem, see the book [24]. The knots in
Examples 3.1, 3.2, and 3.3 are all hyperbolic, and the Volume Conjecture has been verified
for each of them. For 52, this was proved by Ohtsuki [25]; for 61, it was proved by Ohtsuki
and Yokota [27]; and for 72, it follows from Ohtsuki’s work on 7-crossing hyperbolic knots
[26]. It would be interesting to use the formulas given here to independently verify the
Volume Conjecture for these knots.
4 Representing knots as braids
One of the objectives in this paper is to find braid representations of knots that minimize
the number of simple walks. To do that, we will apply several different operations that
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 11
alter the braid word without changing its representative knot. These operations include
reflection, rotation, and cyclic permutation of the braid word, and each of them can be used
to reduce the number of simple walks. We begin with a review of some standard material
on representing knots as braids (see also [5]).
· · ·
β
· · ·
· · ·
α
β
α−1
· · ·
Conjugation
· · ·
β
· · ·
· · ·
β
· · ·
· · ·
β
· · ·
Stabilization
Figure 4: The Markov moves.
Given a braid β, its closure is denoted pβ and is the knot or link obtained by connect-
ing the strands on top with the corresponding strands on bottom without introducing any
additional crossings.
The next result is called Alexander’s theorem and was first proved in 1923, see [1].
Theorem 4.1. Every oriented knot or link is equivalent to the closure pβ for some braid
β P Bm.
Definition 4.2. The Markov moves include conjugation and stabilization (see Figure 4).
Given a braid β P Bm, conjugation involves replacing it with αβα´1 for some α P Bm.
Stabilization involves replacing β with either βσm or βσ´1m . Note that conjugation pre-
serves the braid width and stabilization increases it by one.
The next result is attributed to Markov. For a proof, see [5, Theorem 2.3].
Theorem 4.3. Two braids have equivalent link closures if and only if they are related by a
sequence of Markov moves.
Let β P Bm, and let
SWβ “ tW | W is a simple walk on β with J Ď t2, . . . ,muu (4)
be the set of simple walks on the braid β.
Given a knot, we are interested in finding the braid representative that minimizes the
number of simple walks. Of course, the set of simple walks depends on the braid represen-
tative chosen. In fact, it depends on the braid word since it is not preserved under insertion
(or deletion) of σiσ´1i or σ
´1
i σi into the braid word. This is the analogue, for braids, of the
Reidemeister II move. We explain this important point next.
12 Ars Math. Contemp. 23 (2023) #P1.10
Given a walk on a braid, we say that an arc of the braid is active if it is traversed by a
path of the walk. Similarly, we say that a crossing is active if the walk jumps down from
overcrossing arc to undercrossing arc at that crossing. Thus, the active crossings are the
ones with the local weight ai,˘. In Figures 1, 2, and 3, the active arcs are depicted with
zebra stripes.
Now consider two braid words: γ “ αβ and γ1 “ ασiσ´1i β. (A similar argument
applies to γ1 “ ασ´1i σiβ.) We will show that SWγ Ď SWγ1 .
Suppose W is a simple walk on γ. If both strands i, i ` 1 are active or if they are both
not active, then W extends in a unique way to a simple walk on γ1. If one of the strands
i, i ` 1 is active and the other is not, then W extends to a simple walk on γ1, but possibly
in more than one way. This proves the claim, and in particular, we see that the number of
simple walks is non-decreasing under an elementary insertion.
Recall that a braid word is said to be reduced if it does not contain an occurrence of
σiσ
´1
i or σ
´1
i σi. By the above considerations, for any given knot, we can always assume
that its braid representative is given by a reduced word.
In a similar way, one can show that the set of simple walks is invariant under far com-
mutativity and the Yang-Baxter relation. For far commutativity, this is straightforward, and
we leave the details to the reader. For the Yang-Baxter relation, consider the braid words
γ “ ασiσi`1σiβ and γ1 “ ασi`1σiσi`1β and assume the relevant crossings are j, j ` 1,
and j ` 2.
We claim that any simple walk on γ extends in a unique way to a simple walk on γ1.
There are several cases, depending on which of the three crossings j, j`1, j`2 are active.
If none of the crossings are active, then it extends to a simple walk on γ1. If one of the
crossings is active, and if we make the corresponding crossing on γ1 active, and then it
extends to a simple walk on γ1. If two of the crossings of γ are active, then they must be j
and j`2, and it extends to a simple walk on γ1 again with j and j`2 active crossings. Note
that it is not possible for all three crossings to be active. This shows that |SWγ | “ |SWγ1 |
under the Yang-Baxter relation.
One can also apply the Markov moves to a braid and consider their effect on the set of
simple walks. For instance, under conjugation, one would expect that the resulting braid
word will have a larger set of simple walks. A special case is cyclic permutation, which
involves replacing β “ σε1i1 σ
ε2
i2
¨ ¨ ¨σεℓiℓ with β
1 “ σε2i2 ¨ ¨ ¨σ
εℓ
iℓ
σε1i1 . We will study the effect
of cyclic permutation on the set of simple walks in the next section.
Under stabilization, we will show that the set of simple walks is non-decreasing. Let
β P Bm, β1 “ βσ˘1m P Bm`1, and suppose W is a simple walk on β. If m R J, then
W extends uniquely to a simple walk on β1. If m P J and β1 “ βσm, then we can
extend W to a simple walk on β1 by either making the extra crossing active or by setting
J 1 “ J Y tm ` 1u. If m P J and β1 “ βσ´1m , then we can extend W to a simple walk on
β1 with J 1 “ J Y tm`1u. Notice that for β1 “ βσ´1m , we have one additional simple walk
that does not come from β, namely the one with J “ tm` 1u and the extra crossing made
active. In particular, it follows that |SWβ | ď |SWβ1 |.
Let K be a knot and suppose β P Bm`1 is a braid representative for K with m ě 1.
If β is conjugate to a braid of the form γσ˘m for some braid γ P Bm, then β is said to be
reducible. The braid β is said to be irreducible if it is not reducible.
The next result summarizes our discussion.
Proposition 4.4. If K is a knot, then any braid representative for K that minimizes the
number of simple walks can be assumed to be given by a reduced and irreducible braid word.
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 13
In addition, one can apply symmetry operations to alter the braid word without chang-
ing its knot or link closure. We will use these operations to find braid representatives that
minimize the number of simple walks. The three operations we will consider are called
reflection, rotation, and reversal, and we introduce them next.
For a given braid β, its reflection is denoted β˚ and is obtained by switching all the
crossings of β. If β represents the knot K, then β˚ represents its mirror image K˚. If
β “ σε1i1 ¨ ¨ ¨σ
εℓ
iℓ
, then its reflection is the braid word given by β˚ “ σ´ε1i1 ¨ ¨ ¨σ
´εℓ
iℓ
(see
Figure 5).
For the purposes of computing the colored Jones polynomial, one can use either β or
β˚, since the invariants are related by the simple formula JN,Kpqq “ JN,K˚ pq´1q. The
two braids will have completely different sets of simple walks. In fact, as we shall see
in the next section, the simple walks on β and β˚ are disjoint and complementary to one
another. There is an obvious computational advantage to working with the braid having
fewer simple walks.
In fact, there are other symmetries that can be applied to get a new braid representative
for a knot (or its mirror image). For example, given a braid word β representing a knot, if
one rotates it 1800 in the plane, one obtains a new braid word representing the same knot.
Specifically, if β “ σε1i1 ¨ ¨ ¨σ
εℓ
iℓ
, then the rotated braid word is denoted β: and is given by
β: “ σεℓm´iℓ ¨ ¨ ¨σ
ε1
m´i1 (see Figure 5).
Another example is braid reversal, which is given by reversing the order of the braid
word. Again, the new braid represents the same knot. If β “ σε1i1 ¨ ¨ ¨σ
εℓ
iℓ
, then its reversal
is denoted βr and is given by βr “ σεℓiℓ ¨ ¨ ¨σ
ε1
i1
(see Figure 5). Notice that βr is the braid
obtained from β by rotating it 1800 around a horizontal line in the plane.
There is a one-to-one correspondence between the sets of simple walks on β and βr.
Under the correspondence, the walks have the same set of active crossings, and the weights
for the over- and undercrossings pbi,˘, ci,˘q are switched. In fact, as we shall see, a simple
walk is completely determined by its set of active crossings, and it follows that |SWβ | “
|SWβr |.
β β∗ β† βr
Figure 5: A braid β representing the knot 52 and its reflection β˚, rotation β:, and reversal
βr.
Given a braid word for a knot, applying reflection, rotation, or cyclic permutation will
alter its set of simple walks. Since the computation of the colored Jones polynomial is
exponential in the number of simple walks, it is advantageous to choose the braid represen-
tative that minimizes the number of simple walks.
14 Ars Math. Contemp. 23 (2023) #P1.10
5 Semi-simple walks and cyclic permutation
The main result in this section is an invariance property which asserts that under the cyclic
permutation, the total number of simple walks on a braid β and its reflection β˚ does not
change. To see this, we introduce the notion of semi-simple walks and study their behavior
under reflection.
Recall the definition of SWβ in Equation (4). Previously, we identified walks W with
their weights, given by the ordered product of operators tai,˘, bi,˘, ci,˘u for each crossing
traversed. However, it will be more convenient to record W using only the operators ai,˘,
and we can do so with no loss of information. In the following, we write ai instead of ai,˘;
it is notationally more compact and the sign ˘ can be recovered from the braid word. Thus,
there is a one-to-one correspondence between simple walks on β and (certain) monomials
in ta1, . . . , aℓu, as we shall now explain.
Given a simple walk W , recall that the active crossings are where the walk jumps
down from the overcrossing arc to the undercrossing arc. If the i-th crossing is active, we
record this with ai. As usual, the crossings are labeled 1, 2, . . . , ℓ from top to bottom of
the braid. The collection of active crossings of W determines a monomial in ta1, . . . , aℓu,
and thus we see that a simple walk determines a monomial. Conversely, the monomial in
ta1, . . . , aℓu uniquely determines the simple walk W . We will explain this below, but be-
fore we do, notice that not every monomial corresponds to a simple walk. For example, the
trefoil braid σ31 has three crossings and so there are 2
3 “ 8 possible monomials. However,
it has only one simple walk corresponding to the monomial a2.
Suppose then that ai1 ¨ ¨ ¨ aik is a monomial, indicating that crossings i1, . . . , ik are
active. We perform an oriented smoothing at each active crossing. Since the walk jumps
down there, the crossing type determines which of the arcs are active and which are not.
Specifically, if the crossing is positive, the active arc is the one on the left, and if the
crossing is negative, the active arc is the one on the right (see Figure 6). At each active
crossing, we mark the active arc using some marking scheme. (In all figures, the active
arcs have zebra stripes.) We then extend the marking along the arc through any inactive
crossings and around the back of the braid closure, continuing again through the braid and
around the back as many times as necessary, until reaching another active crossing.
σi σ−1i
Figure 6: An active crossing and its oriented smoothing for σi and σ´1i . The active arc has
zebra stripes and the inactive arc is solid.
One of two things will happen at this active crossing. Either the extended marking is
on an inactive arc, in which case the monomial does not correspond to a valid simple walk,
or it is on the active arc. If the second case holds for all extended arcs, then this is a valid
simple walk. Notice that, in that case, every arc of the partial smoothing of the braid closure
is connected to either an active or an inactive arc. To see that we argue by contradiction. If
the partial smoothing of pβ contains an arc that is not connected to an active or an inactive
arc, then we can follow it along pβ and it will only pass through inactive crossings before
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 15
returning to itself. Therefore, it determines a sublink of the closure of β, which contradicts
the assumption that the closure of β is a knot.
Note that, in the second case, the simple walk is determined by the active or marked
arcs, and once those are specified we can read the full operator by recording all the crossings
it passes through. This explains why the simple walk is determined by its active crossings.
Note that, in order for the marked arcs to be a simple walk, the first strand of the braid at
the top and bottom must be inactive.
This process can alternatively be understood in terms of taking the partial smoothing of
a knot K at a subset of crossings. Given a subset of crossings, we can perform the partial
oriented smoothing of K at the selected crossings. In general, this will produce a link. Our
assumption is that the resulting link can be partitioned into two sublinks, one containing
the active (or marked) arcs the other containing the inactive (or unmarked) arcs.
So, for a given monomial to correspond to a simple walk, it must be the case that the
active and inactive arcs of the braid are contained in different components of the partial
smoothing. Further, it must be the case that the first strand of the braid (at bottom) is
inactive.
We can apply this to understand the behavior of simple walks under taking reflection.
In the mirror image, all the crossings are switched. So, using same monomials to record
simple walk on β and β˚, it follows that taking reflection is equivalent to switching the
active and inactive arcs. That is because each positive crossing of β is negative in β˚ and
vice versa. This is illustrated by switching from left to right or vice versa in Figure 6.
The walks on β and β˚ with the same monomial are dual. We explain this from the
point of view of the full set of weights. Let W be a walk on β with monomial ai1 ¨ ¨ ¨ aik ,
and let W˚ be the corresponding walk on β˚ with monomial ai1 ¨ ¨ ¨ aiℓ . Then W and W˚
trace out disjoint arcs of the braid obtained from β by smoothing the crossings i1, . . . , ik.
In terms of the weights, the walks W and W˚ have the same set of active crossings, but
at the inactive crossings, their local weights are opposite. Specifically, if i is an inactive
crossing and the local weight Wpiq “ bi, then W˚piq “ ci. If instead Wpiq “ ci, then
W˚piq “ bi. Likewise, if Wpiq “ 1, then W
˚
piq “ bici, and if Wpiq “ bici then W
˚
piq “ 1.
Lemma 5.1. The simple walks with J Ď t2, . . . ,mu on a braid β and its reflection β˚ are
disjoint. In other words, SWβ X SWβ˚ “ ∅.
Proof. Suppose W is a simple walk on β with monomial ai1 ¨ ¨ ¨ aik . Then the partial
smoothing of β at the crossings i1, . . . , ik can be partitioned into active and inactive arcs.
Here, we mark the active arcs, with the first strand of the braid at the top inactive. For the
same monomial on the mirror image β˚, the active and inactive arcs will be switched. In
particular, the first strand on β˚ will be active at the top and marked as such. Therefore,
the monomial ai1 ¨ ¨ ¨ aik will not correspond to a valid simple walk on β˚. It follows that
SWβ X SWβ˚ “ ∅, and this completes the proof.
Definition 5.2. Given a braid word β, we say that a walk W on β is semi-simple if it
is a simple walk on β or on β˚ with J Ď t2, . . . ,mu. We use Sβ to denote the set of
semi-simple walks on β. Therefore, Sβ “ SWβ Y SWβ˚ .
Since SWβ and SWβ˚ are disjoint, it follows that |Sβ | “ |SWβ | ` |SWβ˚ |, where
|S| denotes the cardinality of the finite set S.
We leave it as an exercise to show that every monomial ai for 1 ď i ď ℓ corresponds
to a simple walk on either β or β˚. Thus |Sβ | ě n.
16 Ars Math. Contemp. 23 (2023) #P1.10
Theorem 5.3. The set of semi-simple walks Sβ is invariant under cyclic permutation of
the braid word.
The theorem is a direct consequence of the next two lemmas. The first lemma implies
that cyclic permutation of β does not alter the set of simple walks unless β starts with σ1
or σ´11 .
Lemma 5.4. Suppose β “ σε1i1 σ
ε2
i2
¨ ¨ ¨σεℓiℓ is a braid word with i1 ‰ 1. Let β
1 “
σε2i2 ¨ ¨ ¨σ
εℓ
iℓ
σε1i1 be the braid obtained by cyclic permutation. Then SWβ “ SWβ1 .
Proof. Suppose W P SWβ is a simple walk on β with J Ď t2, . . . ,mu. Let W 1 be the
corresponding simple walk on β1, with underlying set J 1. There are three possible cases,
depending on whether i1 and i1 ` 1 lie in J . First, if neither i1 nor i1 ` 1 lie in J , then
cyclic permutation has no effect and W 1 is a simple walk on β1 with J 1 “ J . Second, if
exactly one of i1, i1 ` 1 lies in J , then J 1 ‰ J , but W 1 is nevertheless a simple walk on β1
with J 1 Ď t2, . . . ,mu. Third, if both i1 and i1 ` 1 are in J , then J 1 “ J and W is a valid
simple walk on β1. This completes the proof of the lemma.
The second lemma studies the effect of cyclic permutation for braids that start with σ1
or σ´11 . We will show that cyclic permutation of a braid β has the potential to exchange
simple walks between SWβ and SWβ˚ , but it does not alter the set of semi-simple walks.
Lemma 5.5. Suppose β “ σε1i1 σ
ε2
i2
¨ ¨ ¨σεℓiℓ is a braid word with i1 “ 1. Let β
1 “
σε2i2 ¨ ¨ ¨σ
εℓ
iℓ
σε1i1 be the braid obtained by cyclic permutation. Then Sβ “ Sβ1 .
Proof. Suppose W P SWβ is a simple walk on β with J Ď t2, . . . ,mu. Let W 1 be the
corresponding simple walk on β1, with underlying set J 1. There are two possible cases,
depending on whether or not J contains 2. If 2 R J, then J 1 “ J and so W 1 P SWβ1 .
Similarly, if 2 P J and the monomial for W contains a1 (in which case β necessarily begins
with σ´11 ), then again J
1 “ J and W 1 P SWβ1 . However, if 2 P J and the monomial for
W does not contain a1, then 1 P J 1 and so W 1 R SWβ1 . However, the dual walk pW 1q˚
is simple walk on pβ1q˚ with pJ 1q˚ Ď t2, . . . ,mu, and hence pW 1q˚ P SWpbe1q˚ . In
particular, it follows that the set Sβ “ SWβ Y SWβ˚ of semi-simple walks is unchanged
by cyclic permutation. This completes the proof of the lemma.
The next result states that, up to reordering the crossings, the set of semi-simple walks
on a braid word and its rotation are equal.
Proposition 5.6. Let β “ σε1i1 ¨ ¨ ¨σ
εℓ
iℓ
be a braid word on m strands, and let β: “
σε1m´i1 ¨ ¨ ¨σ
εℓ
m´iℓ be its rotation. Then the sets of semi-simple walks of β and β
: are equal,
namely Sβ “ Sβ: .
Proof. The braid rotation β: is obtained from β by rotating it 1800 in the plane. In order to
relate the semi-simple walks on β and β:, we index the crossings of β: from top to bottom
using n, . . . , 1, and we identify semi-simple walks on β and β: with the subsets of active
crossings. In this way, every semi-simple walk on β and β: corresponds to a monomial
ai1 . . . aik indicating that i1, . . . , ik are the active crossings.
Given a monomial ai1 . . . aik , the semi-simple walk on β is obtained by taking the
smoothing of β at each crossing i1, . . . , ik and locally marking the active and inactive arcs
using a marking scheme that differentiates them. (We mark the active arcs using zebra
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 17
stripes.) Then extend the markings around the braid closure. Since ai1 . . . aik corresponds
to a semi-simple walk, the markings on the active and inactive arcs will not coincide.
The same will be true for β:, provided one follows the same procedures at the corre-
sponding crossings. Since β: is obtained by a 1800 rotation which interchanges the first
and last strands of the braid, this will not preserve SWβ since the new walk may not satisfy
J Ď t2, . . . ,mu. Nevertheless, the semi-simple walks of β and β: are preserved. This
completes the proof.
6 Simple walks on p2, nq torus braids
In this section, we show that the number of simple walks on the braid βn with closure
the p2, nq torus link is given by the n-th term in the Fibonacci sequence 1, 2, 3, 5, 8, 13,
21, 34, 55, 89, 144, . . . We will find a closed form solution for the number of simple walks
and use it to show they grow exponentially in n.
For n ě 1 let βn “ σ´n1 . The closure of βn is the p2, nq torus knot if n is odd and the
p2, nq torus link if n is even. For that reason, we refer to βn as the (negative) p2, nq torus
braid.
Proposition 6.1. Let fpnq be the number of simple walks on the p2, nq torus braid βn.
Then
fpnq “
ˆ
5 `
?
5
10
˙ ˆ
1 `
?
5
2
˙n
`
ˆ
5 ´
?
5
10
˙ ˆ
1 ´
?
5
2
˙n
.
Proof. The first step is to show that the simple walks on βn satisfy the Fibonacci recurrence
relation
fpnq “ fpn ´ 1q ` fpn ´ 2q. (5)
To do this, we establish a bijective correspondence between the set of simple walks on
βn and the union of the sets of simple walks on βn´1 and βn´2. This is accomplished by
extending simple walks on βn´1 and βn´2 to simple walks on βn.
In the following, we identify simple walks with their weights, which we write as mono-
mials in tai, bi, ci | i “ 1, . . . , nu. Notice that all simple walks under consideration will
have J “ t2u.
Given a simple walk w1 on βn´1, set w “ w1an. Then w is a simple walk on βn. See
Figure 7 (left). Since J “ t2u, this is actually the only way to extend w1 to a simple walk
on βn .
Similarly, given a simple walk w1 on βn´2, set w “ w1bn´1cn. Then w is again a
simple walk on βn. See Figure 7 (right). This is the only way to extend w1 to a simple walk
on βn which avoids simple walks extended from βn´1.
 
Figure 7: Extending simple walks from βn´1 and βn´2 to βn.
The two sets of simple walks are disjoint. This can be verified by noting that they
traverse different strands between the pn ´ 1q-st and n-th crossings. Equivalently, one can
18 Ars Math. Contemp. 23 (2023) #P1.10
see this by comparing their weights at the n-th crossing. The simple walks extended from
βn´1 have weight an, whereas those extended from βn´2 have weight cn.
Every simple walk on βn is an extension of one on βn´1 or βn´2. To that end, let w be
a simple walk on βn. Since J “ t2u, at the n-th crossing, either the walk jumps down and
has weight an, or it stays on the overstrand and has weight cn. In the first case, w “ w1an
for a simple walk w1 on βn´1. In the second, w “ w1bn´1cn for some simple walk w1 on
βn´2. This establishes the bijective correspondence, and Equation (5) follows directly.
The second step is to solve the recurrence Relation (5). It is a homogeneous linear
recurrence relation with constant coefficients and characteristic polynomial
pptq “ tn ´ tn´1 ´ tn´2 “ tn´2pt2 ´ t ´ 1q.
This polynomial has two non-zero roots:
t “ 1 ˘
?
5
2
.
Therefore, its general solution is given by fpnq “ c1p 1`
?
5
2 q
n ` c2p 1´
?
5
2 q
n. Using the
values fp1q “ 1 an fp2q “ 2, it follows that
1 “ c1
ˆ
1 `
?
5
2
˙
` c2
ˆ
1 ´
?
5
2
˙
,
2 “ c1
ˆ
1 `
?
5
2
˙2
` c2
ˆ
1 ´
?
5
2
˙2
.
Solving for c1, c2, we find that
c1 “
5 `
?
5
10
and c2 “
5 ´
?
5
10
.
The formula for fpnq follows, and this completes the proof.
7 Simple walks on p3, nq torus braids
In this section, we show that the number of simple walks on the braid γn with closure the
p3, nq torus link is given by the n-th term of the sequence 0, 1, 4, 5, 10, 19, 34, 63, 116, 213,
392, 721, 1326, . . . We will find a closed form solution for the number of simple walks and
use it to show they grow exponentially in n.
For n ě 1, let γn “ pσ´11 σ´12 qn. The closure of γn is the p3, nq torus knot if
n ı 0 pmod 3q and the p3, nq torus link if n ” 0 pmod 3q. For that reason, we refer
to γn as the (negative) p3, nq torus braid.
Proposition 7.1. Let gpnq be the number of simple walks on the p3, nq torus braid γn.
Then
gpnq “ c1αn ` c2βn ` c3γn,
where α, β, γ are the roots of t3 ´ t2 ´ t´ 1 (see Equation (7) for explicit formulas for the
roots) and where
c1 “
1 ` 3α´1
´α2 ` 4α ´ 1 , c2 “
1 ` 3β´1
´β2 ` 4β ´ 1 , c3 “
1 ` 3γ´1
´γ2 ` 4γ ´ 1 .
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 19
Proof. We claim that gpnq satisfies the tribonacci recurrence relation:
gpnq “ gpn ´ 1q ` gpn ´ 2q ` gpn ´ 3q. (6)
The proof of the claim is long, so we first show how to solve the recurrence relation to get
the formula for gpnq.
The tribonacci numbers T pnq are the sequence 0, 1, 1, 2, 4, 7, 13, . . . for n ě 0, and
they also satisfy (6). We will use the closed form solution for T pnq to find a closed form
solution for gpnq. The recurrence Relation (6) has characteristic polynomial pptq “ tn ´
tn´1´tn´2´tn´3 “ tn´3pt3´t2´t´1q. It has one nonzero real root α and two complex
roots β and γ given by
α “ 13 p1 `
3
b
19 ` 3
?
33 ` 3
b
19 ´ 3
?
33q,
β “ 12 p1 ´ α `
a
´3α2 ` 2α ` 5q,
γ “ 12 p1 ´ α ´
a
´3α2 ` 2α ` 5q.
(7)
The closed form solution for T pnq is a linear combination of powers of the roots of the
characteristic polynomial:
T pnq “ α
n
´α2 ` 4α ´ 1 `
βn
´β2 ` 4β ´ 1 `
γn
´γ2 ` 4γ ´ 1 .
The sequence gpnq is related to the tribonacci sequence by the equation
gpnq “ T pnq ` 3T pn ´ 1q.
From this we can write gpnq “ c1αn`c2βn`c3γn and use the values gp1q “ 0, gp2q “
1, gp3q “ 4 to solve for the coefficients c1, c2, c3:
c1 “
1 ` 3α´1
´α2 ` 4α ´ 1 , c2 “
1 ` 3β´1
´β2 ` 4β ´ 1 , c3 “
1 ` 3γ´1
´γ2 ` 4γ ´ 1 .
It remains to prove the claim, namely to show that the simple walks on γn satisfy the
recurrence Relation (6) for all n ě 4. To do this, we establish a bijective correspondence
between the simple walks on γn and the union of the sets of simple walks on γn´1, γn´2
and γn´3. In general, for braids on three strands, the simple walks will have J “ t2u,
J “ t3u or J “ t2, 3u. For the pn, 3q torus braids, all three occur.
We claim that every simple walk on γn´1, γn´2 and γn´3 can be extended to a simple
walk along γn. We prove this by considering the three cases separately. As before, we will
identify simple walks with their weights, which we write as monomials in tai, bi, ci | i “
1, . . . , 2nu.
Suppose w1 is a simple walk on γn´1. If J “ t2u, then it is on the understrand at the
p2n´ 2q-nd crossing, and so w1 “ w2b2n´2. We set w “ w2a2n´2b2n and note that w is a
simple walk on γn with J “ t2u. If J “ t3u, then we set w “ w1a2n. If J “ t2, 3u, then
we set w “ w1b2n ¨ a2n´1c2n. Note that in this last case, the paths become inverted, but
this is allowable for the walks with J “ t2, 3u. Figure 8 shows how the simple walks are
extended. In all three cases it is clear that w is a simple walk on γn.
20 Ars Math. Contemp. 23 (2023) #P1.10
J = {2}
J = {3} J = {2, 3}
 
Figure 8: Extending simple walks from γn´1 to γn.
J = {2} J = {3} J = {2, 3} 
Figure 9: Extending simple walks from γn´2 to γn.
In a similar way, we can extend simple walks on γn´2. Let w1 be a simple walk on
γn´2. If J “ t2u, then we set w “ w1a2n´3c2n´2b2n. If J “ t3u, then we set w “
w1b2n´2a2n´1c2n. If J “ t2, 3u, then we set w “ w1a2n´2b2n ¨ b2n´3c2n´1c2n. In this
last case, notice that the paths become inverted. Figure 9 shows how the simple walks are
extended. In all three cases, it is clear that w is a simple walk on γn.
Lastly, let w1 be a simple walk on γn´3. Then since γn “ γn´3pσ´11 σ´12 q3 with
pσ´11 σ´12 q3 inducing the identity permutation, we can extend w1 to a simple walk on γn
by remaining on the same strands. If J “ t2u, then we set w “ w1b2n´5c2n´3c2n´2b2n.
If J “ t3u, then we set w “ w1b2n´4b2n´3c2n´1c2n. If J “ t2, 3u, then we set w “
w1b2n´5c2n´3c2n´2b2n ¨ b2n´4b2n´3c2n´1c2n. Figure 10 shows how the simple walks are
extended. In all three cases, it is clear that w is a simple walk on γn.
It is not difficult to see that the sets of extended simple walks from γn´1, γn´2, and
γn´3 are all disjoint. For instance, this follows by comparing their weights, and noting
they are pairwise unequal.
The last step is to show that this accounts for all simple walks on γn. We will see that
every simple walk on γn is an extension of a simple walk on γn´1, γn´2 or γn´3.
Suppose w is a simple walk on γn. If J “ t2u, then it must traverse the understrand
at the 2n-th crossing and is on the overstrand just below the p2n ´ 2q-nd crossing. If it
jumps down at the p2n ´ 2q-nd crossing, then w “ w2a2n´2b2n, and w is the extension of
the simple walk w1 “ w2n2n´2 on γn´1. Otherwise, if it remains on the overstrand at the
p2n ´ 2q-nd crossing, then either w “ w1a2n´3c2n´2b2n for w1 a simple walk on γn´2, or
w “ w1b2n´5c2n´3c2n´2b2n for w1 a simple walk on γn´3.
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 21
J = {2} J = {3} J = {2, 3} 
Figure 10: Extending simple walks from γn´3 to γn.
If instead J “ t3u, then it is on the overstrand just below the p2nq-th crossing. If it
jumps down at the p2nq-th crossing, then w “ w1a2n for w1 a simple walk on γn´1. Other-
wise, if it remains on the overstrand at the p2nq-th crossing, then either w “
w1b2n´2a2n´1c2n for w1 a simple walk on γn´2, or w “ w1b2n´4b2n´3c2n´1c2n for w1 a
simple walk on γn´3.
Lastly, suppose J “ t2, 3u. Then w consists of two paths, ends with b2nc2n, and is on
the overstrand below the p2n´ 1q-st crossing. If it jumps down at the p2n´ 1q-st crossing,
then w “ w1b2n ¨ a2n´1c2n for w1 a simple walk on γn´1. If it remains on the overstrand
at the p2n ´ 1q-st crossing, then it is also on the overstrand at the p2n ´ 2q-nd crossing. If
it jumps down at the p2n ´ 2q-nd crossing, then w “ w1a2n´2b2n ¨ b2n´3c2n´1c2n for w1
a simple walk on γn´2. If it remains on the overstrand at the p2n ´ 2q-nd crossing, then it
approaches the p2n ´ 4q-th and p2n ´ 5q-th crossings on understrands, and it follows that
w “ w1b2n´5c2n´3c2n´2b2n ¨ b2n´4b2n´3c2n´1c2n for w1 a simple walk on γn´3.
This shows that every simple walk on γn is obtained by extending a simple walk on
γn´1, γn´2 or γn´3. It follows that there is a bijection correspondence between the set
of simple walks on γn and the union of the simple walks on γn´1, γn´2 and γn´3. The
bijective correspondence implies that the sequence gpnq of simple walks on γn satisfy the
recurrence Relation (6).
8 Minimal braid representatives
In Table 5, we list knots up to 9 crossings with the braid representatives giving minimal
numbers of simple walks. More extensive tables of knots up to 13 crossings and braid
representatives for them can be found online at [7]. (In [7] and Table 1 below, we use the
notation for braid words from sagemath, meaning that a braid word σε1a1 ¨ ¨ ¨σ
εℓ
aℓ
is denoted
by rε1a1, . . . , εℓaℓs. Tables 4 and 5 use even more compactified notation similar to that at
the end of [16].)
These results are empirical. The braid words listed in Table 5 and [7] are the output of
a sagemath program developed by the second author. It takes as input braid representatives
22 Ars Math. Contemp. 23 (2023) #P1.10
Knot Braid Word |SWβ |
11a322 r´1,´1, 2,´3, 4,´3, 2,´3, 4, 1, 2,´3,´2,´2s 51
12a23 r´1,´3, 2,´3,´5, 2, 4, 1, 2,´3,´4, 5, 4,´3,´5, 4, 2s 153
12a155 r´1, 2, 2,´3, 4,´3, 4,´5,´4, 3, 2, 1,´4, 5, 2,´3, 2s 127
12a288 r´1, 2, 2,´3, 2,´3, 2, 1, 2,´3, 4, 2,´3, 4s 71
12a449 r´1, 2, 4,´3, 4, 5, 4, 2,´3,´4,´4,´5, 1, 2, 4,´3, 2s 125
12a494 r´1, 2,´3, 4, 5, 2,´3,´4,´4, 1, 2,´3, 4,´5, 4,´3, 2s 137
12a750 r´1, 2,´3, 2,´3, 5, 4,´3,´4,´4,´5,´4,´4,´3, 1, 4,´2, 3,´2s 183
12n546 r´1, 2, 3, 1,´2, 1, 1, 1, 1,´2,´3,´3, 2s 41
12n601 r´1,´1, 2, 3, 3, 3, 2,´4,´4,´3, 1, 2, 3, 3,´4, 2s 67
12n622 r´1,´1, 2, 2, 3, 3, 2,´4, 1,´2, 3,´4,´2, 3s 47
Table 1: Knots up to 12 crossings whose minimizing braid word begins with σ´11 .
for knots (given by the braids from [19] and [29]) and applies cyclic permutation, reflection,
and rotation. It then selects the braid word that minimizes the number of simple walks.
The output braid word may represent the knot K or its mirror image K˚, whichever has
fewest simple walks.
The braids listed have the fewest simple walks among all known braid representatives
for the given knots. In general, the question of finding a complete list of braid representa-
tives for a given knot is a delicate and open problem. As we shall see, it is not enough to
consider only braid representatives of minimal width. Even if it were, it is an open problem
to develop an algorithm for computing the braid width of a knot (see Open Problem 1 in
[6]). Nevertheless, these problems have been studied extensively, and much is known about
minimal braid representatives of low-crossing knots; see [13, 16] and [29].
Given a knot, one can look for braid representatives that minimize its braid width or the
braid length. For many knots, there is a braid representative that simultaneously minimizes
both the width and length, but in general, the braid representatives that minimize width
need not be the same as the ones that minimize length. The earliest known examples are
the knots 16472381 and 161223549, which were discovered by Stoimenow and have braid
width 4 but no minimal length braid representative of width 4, [28, Figure 7].
This interesting aspect has been further studied by Gittings [13] and Van Cott [30], and
the “smallest” example is the knot 10136. For all other knots with up to 10 crossings, there
is a braid representative that simultaneously minimizes the braid width and length. Further
examples of knots whose minimal width braid representatives are not minimal length are
listed in Table 4. (These examples come from [19].)
Interestingly, the braid representative that minimizes the number of simple walks is not
always a minimal length braid, nor is it always a minimal width braid either. For example,
consider the knots 10136 and 11n8 and their braid representatives in Table 4. For 10136, the
number of simple walks is minimized on a braid representative of minimal length but not
one of minimal width, whereas for 11n8, the number of simple walks is minimized on a
braid representative of minimal width but not one of minimal length. Similar examples can
be found among the other knots listed in Table 4.
Our computations suggest that, for any knot, one can always minimize the number
of simple walks on a braid representative of minimal width or minimal length. This is
an interesting problem for future investigation. In order to make progress, we need more
information about the minimal width and minimal length braid representatives for knots.
At present, we do not have complete information on the 13-crossing knots. In particular,
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 23
we do not know which 13-crossing knots have minimal length braid representatives that
are not minimal width. The braid representatives for the 13-crossing knots from [29] are
known to be of minimal width, but they are not known to be of minimal length.
Notice that for every knot in Table 5, the braid representative that minimizes the number
of simple walks begins with σ1. This is actually true for all knots up to 10 crossings, but
not immediately true for knots with 11 or more crossings (see [7]).
In general, by Lemma 5.4, the minimizing braid representative can always be chosen to
begin with either σ1 or σ´11 . For knots with 11 and 12 crossings, there are only a handful
of examples whose minimizing braid representative begins with σ´11 and not σ1. They are
listed in Table 1. (There are in addition 82 examples among the knots with 13 crossings,
see [7].) In each case, we can find a minimizing braid representative that begins with σ1 by
reversing the braid word and applying cyclic permutation. We explain these steps in more
detail.
Take, for example, the first knot in Table 1, namely 11a322. Its minimizing braid rep-
resentative is the braid word σ´21 σ2σ
´1
3 σ4σ
´1
3 σ2σ
´1
3 σ4σ1σ2σ
´1
3 σ
´2
2 . The reversed braid
word will have the same number of simple walks, so it follows that σ´22 σ3σ2σ1
σ4σ
´1
3 σ2σ
´1
3 σ4σ
´1
3 σ2σ
´2
1 is also a minimizing braid word for 11a322. Now repeated ap-
plication of Lemma 5.4 shows that the braid word σ1σ4σ´13 σ2σ
´1
3 σ4σ
´1
3 σ2σ
´2
1 σ
´2
2 σ3σ2
is also minimizing for 11a322.
3 4 5 6 7 8 9 10 11 12 13
0
20
40
60 Average Number of Simple Walks
Table 2: Average number of simple walks by crossing number.
The same method applies to the other knots in Table 1. Each one admits a minimizing
braid word that starts with σ1. A similar argument applies to the braid representatives for
the 13 crossing knots that begin with σ´11 . This follows by a routine but somewhat tedious
exercise.
Table 2 shows the growth rate of the number of simple walks as a function of the
crossing number of the knot. Table 3 shows the growth rate of the number of simple walks
as a function of the braid length. Note that Table 3 contains information for knots with up
to 12 crossings but not the 13-crossing knots. The reason is that we do not have definitive
information about the braid representatives of minimal length for the 13-crossing knots.
We end this paper with a few questions and open problems for future investigation. One
is whether braid words that minimize the number of simple walks have a preferred shape
or form. By Proposition 4.4, we can assume the braid word is reduced and irreducible, and
by Lemma 5.4, we can assume it begins with σ1 or σ´11 . We conjecture the braid word
can always be chosen to begin with σ1. Further results as to the shape of minimizing braid
words would be helpful for developing efficient search algorithms.
24 Ars Math. Contemp. 23 (2023) #P1.10
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
0
50
100 Average Number of Simple Walks
Table 3: Average number of simple walks by braid length.
More generally, it would be extremely useful to automate the generation of minimal
width and/or minimal length braid representatives for a given knot. Such tools would allow
fast computation of the colored Jones polynomial and other quantum knot invariants, en-
abling calculations for higher crossing knots, including those in the knot tables of Burton
[8], who has recently extended the classification of knots to 19 crossings.
Auxilliary files
Sagemath programs and datasets are available online [7]. This includes a program that
generates all the simple walks as operators for a given braid and another that selects braid
words that minimize the number of simple walks. It also includes input datasets used to
create Tables 4 and 5, as well as output datasets of braid words that minimize the number
of simple walks for knots up to 13 crossings.
Knot Minimal width braid |SW| Minimal length braid |SW|
10136 123´121´1223´12´21 21 12´13´12243´1412´1 17
11n8 123´12´112´11´1232221 20 121´1213´12243´141 23
11n121 12322´112´11´1223´121 20 12123´121´1243´141 21
11n131 12322´2122´13´1212´1 17 123´121322´13´143´14 21
12n17 2´112´134´232´23´143´141 63 p12´1q23´1224325´145´1 52
12n20 3´112´1p32´1q21´123´1221 26 p12´1q2p3´12q243´141 32
12n24 12´132´232´13´11´123´121 35 2´112´13´12243´341 32
12n65 2´1123´14243´123´14´123´11 26 121´13´123´14´13254´151 25
12n119 2´1122´1313p12´1q23´11 31 2´113´1243´14122´112 23
12n284 p12´2q23132´112´13´1 36 2´112´13´123´143´123´141 32
12n311 4121´123´142´13´1432´13´11 37 2´11324´13´123´154´1351 30
12n314 2´13´1p12´1q33132´11 34 p12´1q23´12143´12´143 24
12n358 2´13´1121´1212´13132´11 24 3´12´11432´134´13421 20
12n362 123´221´1223´12´132´11 42 3´12´112´243´141221 24
12n403 12´1342´13´11´123´121 37 1213´12´112´143´1241 20
12n482 2´13´113´1221´2232´112 29 2´112´13´1243´123´241 32
Table 4: Simple walks for representatives of minimal braid width and length.
H. U. Boden and M. Shimoda: Braid representatives minimizing the number of simple walks 25
Knot Braid Word |SW|
31 13 1
41 p12´1q2 2
51 15 3
52 12312´1 2
61 121´13´123´11 3
62 132´112´1 4
63 122´112´2 5
71 17 8
72 12331´1213´1 3
73 1421´121 6
74 12232´13212´1 5
75 1321´1221 5
76 122´1132´13 8
77 p12´1q232´13 8
81 1234´12´13212´14´1 4
82 152´112´1 9
83 121´13´123´14´134´11 9
84 123´123´312´1 7
85 p132´1q2 8
86 1321´13´123´11 7
87 142´112´2 10
88 1221´13´123´21 10
89 132´112´3 9
810 132´1122´2 9
811 121´1223´123´11 7
812 p12´134´1q2 14
813 1223´123´212´1 8
814 1221´1p23´1q21 8
815 123132´132 9
816 122´112´1122´1 9
817 1p12´1q32´1 9
818 p12´1q4 10
819 p123q2 5
820 12312´3 5
821 1221´2221 6
91 19 21
92 12343´142´132212´1 5
93 1621´121 14
94 1232243´112 9
95 121´12342´13´14321 10
96 1521´1221 12
97 1234223´112´1 9
Knot Braid Word |SW|
98 1412´13´1423´112´1 13
99 1231´1214 12
910 1232´13231´121 11
911 12´113312´13 15
912 123´141´123´1414 13
913 1232´1321´1231 10
914 121´13´123´143´141 11
915 1223´141´123´141 16
916 1231´12213 10
917 p12´1q22´1p32´1q2 17
918 122322312´13´1 10
919 12´23´1243´1412´1 14
920 132´13132´13 15
921 121´123´1243´141 12
922 12´1312´332´1 17
923 1221´123´122132 15
924 1312´1312´3 17
925 12334´112´134´1 14
926 12´112312´132´1 13
927 122´112´232´13 15
928 12´11312´232 17
929 p12´132´1q22´1 16
930 122´212´132´13 16
931 12´11312´1322´1 15
932 1p12´1q2312´13 14
933 p12´1q22´1312´13 16
934 12´13p12´1q232´1 13
935 1234´1341´142´132123´1 17
936 132´11312´13 14
937 p12´13q243´123´12´14´1 29
938 123221´123´1221 13
939 123´12143´12´14´1342 18
940 12´1312´1132´13 14
941 13´1423´123´212´134 20
942 123´1212´33´1 7
943 1233´1123´12 8
944 123´12212´13´12´1 7
945 123312´132´1 8
946 123´1212´132´13 8
947 p123´12q23´1 8
948 12322´112´13´11´121 9
949 1221312´13212´1 9
Table 5: Knots up to 9 crossings and braid words minimizing the number of simple walks.
26 Ars Math. Contemp. 23 (2023) #P1.10
ORCID iDs
Hans U. Boden https://orcid.org/0000-0001-5516-8327
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