Cobordism of algebraic knots defined by Brieskorn polynomials, II

TL;DR

This paper advances the understanding of algebraic knot cobordisms related to Brieskorn polynomials by establishing new invariants, relations, and properties, including the infinite order of certain spherical algebraic knots in the cobordism group.

Contribution

It introduces new results on Fox--Milnor relations, algebraic cobordism decomposition, and cyclic suspensions, extending previous work on cobordisms of algebraic knots.

Findings

Exponents are cobordism invariants under certain conditions.

A spherical algebraic knot has infinite order in the cobordism group.

New relations and decompositions for algebraic cobordism classes.

Abstract

In our previous paper, we obtained several results concerning cobordisms of algebraic knots associated with Brieskorn polynomials: for example, under certain conditions, we showed that the exponents are cobordism invariants. In this paper, we further obtain new results concerning the Fox--Milnor type relations, decomposition of the algebraic cobordism class of an algebraic knot associated with a Brieskorn polynomial that has a null-cobordant factor over the field of rational numbers, and cyclic suspensions of knots. As a corollary, we show that a spherical algebraic knot associated with a Brieskorn polynomial has infinite order in the knot cobordism group.