Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Infinite families of non-fibered twisted torus knots

Adnan, Kyungbae Park

Abstract

We present explicit infinite families of twisted torus knots that are not fibered. Our approach relies on an explicit formula for the Alexander polynomial derived in our previous work. We show that the leading coefficients of the Alexander polynomials of twisted torus knots can take arbitrary integer values, which immediately yields infinitely many examples of non-fibered twisted torus knots.

Infinite families of non-fibered twisted torus knots

Abstract

We present explicit infinite families of twisted torus knots that are not fibered. Our approach relies on an explicit formula for the Alexander polynomial derived in our previous work. We show that the leading coefficients of the Alexander polynomials of twisted torus knots can take arbitrary integer values, which immediately yields infinitely many examples of non-fibered twisted torus knots.
Paper Structure (5 sections, 5 theorems, 45 equations, 1 figure)

This paper contains 5 sections, 5 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Alexander polynomial of twisted torus knots
  3. Leading coefficients and degrees of certain twisted torus knots
  4. Proof of Theorem \ref{['thm:main-1']}
  5. Proof of Theorem \ref{['thm:main-2']}

Key Result

Theorem 1.1

Let $r > 0$ and $s < -1$. Then the Alexander polynomial of the twisted torus knot has leading coefficient $r$ and degree $r|s|(r|s| - r - 2) + 2$.

Figures (1)

  • Figure 1: The twisted torus knot $T(10,3;5,-1)$, which is not fibered.

Theorems & Definitions (6)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Conjecture 1.6