Branched twist spins and knot determinants

TL;DR

This paper introduces a method to distinguish non-equivalent branched twist spins using knot determinants, supported by a fundamental group presentation and algebraic calculations.

Contribution

It provides a sufficient condition based on knot determinants to differentiate non-trivial branched twist spins, extending previous presentations of their fundamental groups.

Findings

A presentation of the fundamental group of branched twist spins' complements.

Calculation of first elementary ideals for these knots.

A criterion using knot determinants to distinguish non-equivalent spins.

Abstract

A branched twist spin is a generalization of twist spun knots, which appeared in the study of locally smooth circle actions on the $4$-sphere due to Montgomery, Yang, Fintushel and Pao. In this paper, we give a sufficient condition to distinguish non-equivalent, non-trivial branched twist spins by using knot determinants. To prove the assertion, we give a presentation of the fundamental group of the complement of a branched twist spin, which generalizes a presentation of Plotnick, calculate the first elementary ideals and obtain the condition of the knot determinants by substituting $-1$ for the indeterminate.