Critical point counts in knot cobordisms: abelian and metacyclic invariants

TL;DR

This paper investigates the relationship between knot cobordisms and critical point counts using homological invariants from branched covers, providing obstructions and examples that deepen understanding of knot transformations.

Contribution

It introduces new obstructions based on cyclic and metacyclic branched covers to determine possible critical point configurations in knot cobordisms.

Findings

Obstructions to critical point counts are derived from homological invariants.

Examples demonstrate the existence of knots with prescribed critical point constraints.

A specific knot example shows minimal critical points in cobordisms between a knot and its reverse.

Abstract

For pairs of knots K and J in the three-sphere, we consider the set of four-tuples of integers (g,x,y,z) for which there is a cobordism from K to J of genus g having x, y, and z, critical points of index 0, 1, and 2, respectively. We describe basic properties that such sets must satisfy and then build obstructions to membership in the set. These obstructions are based on homological invariants arising from cyclic and metacyclic branched covering spaces. A series of examples is presented. A concluding example demonstrates that for each pair of integers g and n, there exists a ribbon knot K for which any genus g cobordism from K to its reverse K^r must have at least n critical points of each index.