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Data for Twisted Alexander polynomials for hyperbolic knots
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Dunfield, Nathan, Friedl, Stefan and Jackson, Nicholas (2021) Data for Twisted Alexander polynomials for hyperbolic knots. [Dataset]
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Official URL: https://doi.org/10.7910/DVN/OK6YGC
Abstract
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.
Item Type: | Dataset | ||||||
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Alternative Title: | Twisted Alexander polynomials for hyperbolic knots : data and software | ||||||
Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Type of Data: | Modelling data | ||||||
Library of Congress Subject Headings (LCSH): | Polynomials, Knot theory, Knot polynomials | ||||||
Publisher: | University of Warwick, Mathematics Institute | ||||||
Official Date: | 10 March 2021 | ||||||
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Status: | Not Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||
Copyright Holders: | University of Warwick | ||||||
Description: | Data record consists of two gzip archives and an accompanying PDF readme file. |
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