Cappell-Shaneson knot pairs with the same Alexander polynomial

TL;DR

This paper constructs an infinite family of Cappell-Shaneson knot pairs, including examples with identical Alexander polynomials that are nonetheless inequivalent, advancing understanding of knot pair classifications.

Contribution

It introduces new examples of Cappell-Shaneson knot pairs, including those with the same Alexander polynomial but different knot types.

Findings

Constructed an infinite family of Cappell-Shaneson knot pairs.

Provided examples of inequivalent knots sharing the same Alexander polynomial.

Abstract

It is well known that for $m\geq 2$ there are at most two non-equivalent $m$-knots with diffeomorphic exterior. Such pair of knots will be called $\textit{ non-reflexive knot pair}$. A classical problem in topology is to determine all dimensions where such knot pairs exist. In 1976 Cappell and Shaneson gave a method of constructing non-reflexive knot pairs. In the present paper we construct an infinite family of new examples of Cappell-Shaneson knot pairs, and give examples of Cappell-Shaneson knot pairs that have the same Alexander polynomial but are inequivalent.