A Refinement of the Spanning Surface Defect in $3$ and $4$ Dimensions

TL;DR

This paper refines the spanning surface defect for knots, extending it to four-dimensional surfaces, and compares 3- and 4-dimensional settings to improve bounds on non-orientable 4-genus and establish a connected sum formula.

Contribution

It introduces a refined invariant for knots that incorporates 4-dimensional surfaces and provides new comparisons and formulas linking 3- and 4-dimensional knot invariants.

Findings

Refined the spanning surface defect to include 4-dimensional surfaces.

Established a connected sum formula for the new invariant.

Reframed non-orientable slice-torus bounds on non-orientable 4-genus.

Abstract

The spanning surface defect uses spanning surfaces of a knot in the $3$-sphere to measure how far a knot is from being alternating. We refine the spanning surface defect and extend the definition to take into account surfaces in the $4$-ball. We use these extensions to make comparisons between the $3$- and $4$-dimensional settings, to reframe non-orientable slice-torus bounds on the non-orientable $4$-genus, and to prove a connected sum formula.