Unknotting numbers and triple point cancelling numbers of torus-covering knots

TL;DR

This paper investigates the minimal number of 1-handle additions needed to unknot or simplify torus-covering knots, providing bounds using combinatorial and algebraic invariants.

Contribution

It introduces bounds for unknotting and triple point cancelling numbers of torus-covering knots using m-charts, quandle colorings, and cocycle invariants.

Findings

Upper bounds via m-charts on the torus

Lower bounds using quandle colorings

Quantitative bounds for specific torus-covering knots

Abstract

It is known that any surface knot can be transformed to an unknotted surface knot or a surface knot which has a diagram with no triple points by a finite number of 1-handle additions. The minimum number of such 1-handles is called the unknotting number or the triple point cancelling number, respectively. In this paper, we give upper bounds and lower bounds of unknotting numbers and triple point cancelling numbers of torus-covering knots, which are surface knots in the form of coverings over the standard torus $T$. Upper bounds are given by using $m$-charts on $T$ presenting torus-covering knots, and lower bounds are given by using quandle colorings and quandle cocycle invariants.