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
*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/
ISSN 1855-3966 (tiskana izd.), ISSN 1855-3974 (elektronska izd.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.10
https://doi.org/10.26493/1855-3974.2730.6ac
(Dostopno tudi na http://amc-journal.eu)
Predstavniki pletenic, ki minimizirajo število
enostavnih sprehodov*
Hans U. Boden † , Matthew Shimoda ‡
Mathematics & Statistics, McMaster University, Hamilton, Ontario, Canada
Prejeto 15. novembra 2021, sprejeto 19. julija 2022, objavljeno na spletu 21. novembra 2022
Povzetek
V članku razvijemo metode za določitev predstavnika pletenic, ki minimizira število
enostavnih sprehodov v danem vozlu. Takšne pletenice vodijo k učinkoviti metodi za
izračun barvnega Jonesovega polinoma vozla, s čimer sledimo pristopu, ki ga je razvil Ar-
mond, implementirala pa sta ga Hajij in Levitt. To metodo uporabimo za izračun barvnega
Jonesovega polinoma v sklenjeni obliki za vozle 52, 61, in 72. Množica enostavnih vozlov
enostavnih sprehodov se lahko spremeni z zrcaljenjem, rotacijo in ciklično permutacijo
pletenice; dokažemo invariantno lastnost, ki povezuje enostavne sprehode pletenice s tis-
timi, ki pripadajo njeni zrcalni obliki pri ciklični permutaciji. Preučujemo stopnjo rasti
števila enostavnih sprehodov za družine torusnih vozlov. Nazadnje predstavimo tabelo
pleteničnih besed, ki minimizirajo število enostavnih sprehodov za vozle z do 13 križišči.
Ključne besede: Vozli, pletenice, enostaven sprehod, barvni Jonesov polinom.
Math. Subj. Class. (2020): 57K10, 57K14
*Avtorja sta še posebej hvaležna Alexandru Stoimenowu za njegov ključni prispevek. Rada bi se zahvalila tudi
Homayunu Karimi, Robertu Osburnu, Andrewu Nicasu, Willu Rushworthu in Corneliji Van Cott za dragoceno
povratno informacijo.
†Kontaktni avtor. Avtor je bil financiran s strani Natural Sciences and Engineering Research Council of
Canada.
‡Avtor je bil financiran z nagrado USRA in s Stewartovo nagrado, ki jo podeljuje McMaster University.
E-poštna naslova: boden@mcmaster.ca (Hans U. Boden), mattshimoda@hotmail.com (Matthew Shimoda)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/