An obstruction to a knot being deform-spun via Alexander polynomials

TL;DR

This paper investigates the limitations of deform-spinning in knot theory, demonstrating that Alexander polynomial constraints prevent certain co-dimension two knots in spheres from being generated through deform-spinning from lower-dimensional knots.

Contribution

It establishes new algebraic constraints on Alexander polynomials that obstruct some knots from being obtained via deform-spinning, revealing limitations of this construction.

Findings

Not all co-dimension 2 knots in S^n are deform-spun from knots in S^{n-1}.

Alexander polynomial constraints serve as obstructions to deform-spinning.

The paper provides conditions that restrict the deform-spinning process in knot theory.

Abstract

We show that if a co-dimension two knot is deform-spun from a lower-dimensional co-dimension 2 knot, there are constraints on the Alexander polynomials. In particular this shows, for all n, that not all co-dimension 2 knots in S^n are deform-spun from knots in S^{n-1}.