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 -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 -ball. We use these extensions to make comparisons between the - and -dimensional settings, to reframe non-orientable slice-torus bounds on the non-orientable -genus, and to prove a connected sum formula.
Cite
@article{arxiv.2602.13186,
title = {A Refinement of the Spanning Surface Defect in $3$ and $4$ Dimensions},
author = {Julia Knihs and Jeanette Patel and Joshua M. Sabloff and Thea Rugg},
journal= {arXiv preprint arXiv:2602.13186},
year = {2026}
}
Comments
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