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Twisted Alexander polynomials of knots associated to the regular representations of finite groups

Takayuki Morifuji, Masaaki Suzuki

Abstract

The twisted Alexander polynomial of a knot is defined associated to a linear representation of the knot group. If there exists a surjective homomorphism of a knot group onto a finite group, then we obtain a representation of the knot group by the composition of the surjective homomorphism and the regular representation of the finite group. In this paper, we provide several formulas of the twisted Alexander polynomial of a knot associated to such representations in terms of the Alexander polynomial.

Twisted Alexander polynomials of knots associated to the regular representations of finite groups

Abstract

The twisted Alexander polynomial of a knot is defined associated to a linear representation of the knot group. If there exists a surjective homomorphism of a knot group onto a finite group, then we obtain a representation of the knot group by the composition of the surjective homomorphism and the regular representation of the finite group. In this paper, we provide several formulas of the twisted Alexander polynomial of a knot associated to such representations in terms of the Alexander polynomial.
Paper Structure (6 sections, 16 theorems, 77 equations, 5 tables)

This paper contains 6 sections, 16 theorems, 77 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Twisted Alexander polynomials
  3. Dihedral groups $D_n$
  4. Metacyclic groups $G(m,p|k)$
  5. Dicyclic groups $\hbox{Dic}_n$
  6. Concluding remarks

Key Result

Lemma 2.2

Theorems & Definitions (28)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Lemma 3.1
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • ...and 18 more