Squeezed Knots

TL;DR

This paper introduces the class of squeezed knots, explores their properties, and discusses how quantum invariants, especially refined Rasmussen invariants, can be used to obstruct their squeezedness.

Contribution

It defines squeezed knots via cobordisms between torus knots and demonstrates how quantum invariants can obstruct a knot's squeezedness, expanding understanding of knot complexity.

Findings

Squeezed knots form a large class of knots.

Quantum invariants can effectively obstruct squeezedness.

Refinements of Rasmussen invariant are useful in this context.

Abstract

Squeezed knots are those knots that appear as slices of genus-minimizing oriented smooth cobordisms between positive and negative torus knots. We show that this class of knots is large and discuss how to obstruct squeezedness. The most effective obstructions appear to come from quantum knot invariants, notably including refinements of the Rasmussen invariant due to Lipshitz-Sarkar and Sarkar-Scaduto-Stoffregen involving stable cohomology operations on Khovanov homology.