The Complexity of Shake Slice Knots

TL;DR

This paper introduces a new complexity measure for shake-slice knots, demonstrating that for any framing and complexity level, such knots exist, and explores their properties using dualizable patterns and signature invariants.

Contribution

It defines a novel complexity concept for shake-slice knots and constructs examples with arbitrarily high complexity, utilizing dualizable patterns and signature analysis.

Findings

Existence of shake-slice knots with arbitrary complexity levels.

Bounded complexity through signature behavior under twisting.

Use of dualizable patterns to construct and analyze knots.

Abstract

We define a notion of complexity for shake-slice knots which is analogous to the definition of complexity for h-cobordisms studied by Morgan-Szab\'o. We prove that for each framing $n \ne 0$ and complexity $c \ge 0$, there is an $n$-shake-slice knot with complexity at least $c$. Our construction makes use of dualizable patterns, and we include a crash course in their properties. We bound complexity by studying the behavior of the classical knot signature and the Levine-Tristram signature of a knot under the operation of twisting algebraically-one strands.