Abstract
We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3sphere via a lift of the holonomy representation to. 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 313209 hyperbolic knots in S ^{3} 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 X 0 of thecharacter 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 X 0. We use this to help explain some of the patterns observed for knots in S ^{3}, and explore a potential relationship between this universal polynomial and the CullerShalen theory of surfaces associated to ideal points.
Original language  English (US) 

Pages (fromto)  329352 
Number of pages  24 
Journal  Experimental Mathematics 
Volume  21 
Issue number  4 
DOIs  
State  Published  Dec 1 2012 
Keywords
 character varieties
 genus bounds
 hyperbolic knots
 twisted Alexander polynomial
 twisted torsion
ASJC Scopus subject areas
 Mathematics(all)
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Twisted Alexander polynomials for hyperbolic knots: data and software
Dunfield, N. M. (Creator), Friedl, S. (Creator) & Jackson, N. (Creator), Harvard Dataverse, Oct 31 2020
DOI: 10.7910/DVN/OK6YGC
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