The Complexity of Shake Slice 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-Szabó. 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.

Revised to include a sharper statement, and give more insight into the construction of the examples. Comments welcome!

References in corpus (2)

• Knot traces and concordance
• Shake slice and shake concordant knots