Cappell-Shaneson polynomials

TL;DR

This paper studies Cappell-Shaneson polynomials, which are characteristic polynomials of special matrices related to embeddings of spheres, providing classifications for degrees 4 and 5, and constructing series for degree 6.

Contribution

It offers a new algebraic interpretation of Cappell-Shaneson polynomials and classifies all such polynomials of degrees 4 and 5, with new series for degree 6.

Findings

Complete lists of Cappell-Shaneson polynomials of degrees 4 and 5.

Construction of infinite series of degree 6 Cappell-Shaneson polynomials.

Abstract

S. Cappell and J. Shaneson constructed a pair of inequivalent embeddings of $(n-1)$-spheres in homotopy $(n+1)$-spheres for every square matrix of order $n$ with special properties (a Cappell-Shaneson matrix). A Cappell-Shaneson polynomial is the characteristic polynomial of a Cappell-Shaneson matrix. In this paper, we interpret part of the definition of Cappell-Shaneson polynomial as algebraic conditions on polynomials in terms of signed reciprocal polynomial and reduction modulo primes, and give complete lists of all Cappell-Shaneson polynomials of degrees $4$ and $5$. We construct several infinite series of Cappell-Shaneson polynomials of degree $6$.