The effect of infecting curves on knot concordance

TL;DR

This paper introduces a new method to obstruct knot concordance, demonstrating that infinitely many distinct concordance classes can be constructed by varying infecting curves, even with fixed base knots and doubling operators.

Contribution

The paper presents a novel approach to obstruct knot concordance by varying infecting curves, expanding the understanding of concordance class distinctions.

Findings

Infinitely many concordance classes can be constructed by varying infecting curves.

Distinct classes are found even with fixed base knots and doubling operators.

The new method complements existing obstructions like Casson-Gordon invariants.

Abstract

Various obstructions to knot concordance have been found using Casson-Gordon invariants, higher-order Alexander polynomials, as well as von-Neumann rho-invariants. Examples have been produced using (iterated) doubling operations K=R(c,J), and considering these as parametrized by invariants of the base knot J and doubling operator R. In this paper, we introduce a new mew method to obstruct concordance. We show that infinitely many distinct concordance classes may be constructed by varying the infecting curve c in S^3-R. Distinct concordance classes are found even while fixing the base knot, the doubling operator, and the order of c in the Alexander module.