2026-10-14T13:13:04Z https://api.figshare.com/v2/oai

oai:figshare.com:article/33820678 2026-09-15T19:57:21Z portal_1197 item_type_3 month_year_09_2026

Data for Twisted Alexander polynomials for hyperbolic knots Nathan Dunfield (24954334) Stefan Friedl (24759520) Nicholas Jackson (3148170) Polynomials Knot theory Knot polynomials Migrated from ePrints We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to SL(2,C). It is an unambiguous symmetric Laurent polynomial whose coefficients lie in the trace field of the knot. It contains information about genus, fibering, and chirality, and moreover, is powerful enough to sometimes detect mutation. We calculated this invariant numerically for all 313 209 hyperbolic knots in S3 with at most 15 crossings, and found that in all cases it gave a sharp bound on the genus of the knot and determined both fibering and chirality. We also study how such twisted Alexander polynomials vary as one moves around in an irreducible component X0 of the SL(2,C)-character variety of the knot group. We show how to understand all of these polynomials at once in terms of a polynomial whose coefficients lie in the function field of X0. We use this to help explain some of the patterns observed for knots in S3, and explore a potential relationship between this universal polynomial and the Culler–Shalen theory of surfaces associated to ideal points.<br><br>Data record consists of two gzip archives and an accompanying PDF readme file. This is the software and data to accompany the paper "Twisted Alexander polynomials of hyperbolic knots" see the links below. It includes A table the hyperbolic torsion polynomial for all hyperbolic knots with at most 15 crossings. Software to compute the hyperbolic torsion polynomial for any knot. This part of the code is now incorporated into SnapPy. Software for trying to compute the genus of a knot and checking whether it fibers. See the "readme.pdf" for more details. 2026-09-15T19:57:21Z Dataset Dataset 10.82444/warw.33820678.v1 https://figshare.com/articles/dataset/Data_for_Twisted_Alexander_polynomials_for_hyperbolic_knots/33820678 CC BY 4.0