Twisted Alexander polynomials of 2-bridge knots
Jim Hoste (Pitzer College), Patrick D. Shanahan (Loyola Marymount, University)
TL;DR
This paper derives formulas for twisted Alexander polynomials of 2-bridge knots, including torus and genus-one knots, confirming a conjecture for these families.
Contribution
It provides explicit formulas for twisted Alexander polynomials of various 2-bridge knots and verifies a conjecture by Hirasawa and Murasugi.
Findings
Formulas for twisted Alexander polynomials of specific 2-bridge knots
Confirmation of Hirasawa and Murasugi's conjecture for these knots
Application to families like torus and genus-one knots
Abstract
We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted Alexander polynomials. We use these formulae to confirm a conjecture of Hirasawa and Murasugi for these knots.
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Advanced Combinatorial Mathematics