Notes on constructions of knots with the same trace

TL;DR

This paper explains a technique for constructing infinitely many knots with identical $m$-traces using dualizable patterns, connecting it to previous methods and showing its application to knots with the same 4-surgery.

Contribution

It demonstrates that the $(ullet m)$ operation can be understood via Gompf and Miyazaki's dualizable patterns, unifying different knot construction approaches.

Findings

The $(ullet m)$ operation aligns with dualizable pattern techniques.

It explains the family of knots with the same 4-surgery as described by Teragaito.

The technique provides a unified framework for constructing knots with identical $m$-traces.

Abstract

The $m$-trace of a knot is the $4$-manifold obtained from $\mathbf{B}^4$ by attaching a $2$-handle along the knot with $m$-framing. In 2015, Abe, Jong, Luecke and Osoinach introduced a technique to construct infinitely many knots with the same $m$-trace, which is called the operation $(\ast m)$. In this paper, we prove that their technique can be explained in terms of Gompf and Miyazaki's dualizable pattern. In addition, we show that the family of knots admitting the same $4$-surgery given by Teragaito can be explained by the operation $(\ast m)$.