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

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.

29 pages, 18 figures. The previous version did not take into account Ito's work on the 3-dimensional spanning surface defect. The main definitions now build off of those of Ito, and the old results 1.3 - 1.6 have been properly attributed. As a result, the old Section 4 has been removed; the focus of the new version is on comparisons between 3- and 4-dimensional phenomena