Smooth and Topological Nonorientable Four-Genera of Low-Crossing Knots
Abstract
We compare linking-form bounds with the smooth and locally flat nonorientable 4-genera of low-crossing knots. For every prime knot with at most 13 crossings whose Witt-level linking-form bound is at least 2, this bound equals both genera. We also exhibit 41 prime knots with 12 or 13 crossings that bound locally flat Möbius bands but no smooth ones, and whose smooth nonorientable 4-genus is 2. Our methods combine linking forms, Heegaard Floer correction terms and definite-lattice obstructions with explicit band constructions. They determine 3,300 of the 3,525 values left unknown in KnotInfo on 2 November 2026; the remaining 225 knots have range [1,2]. We obtain new lower bounds of 2 for 2,102 knots with 14 crossings and correct previously reported values for three knots with 11 crossings.
Related
- Linking-form splitting and four-dimensional genera of knots
- The Crossing Number and Arc Index of Satellite Knots
- Examples of descendent generating series for Pandharipande--Thomas stable pairs on smooth projective Fano threefolds via one-dimensional wall-crossing
- The non-perturbative topological string: from resurgence to wall-crossing of DT invariants
- On smooth and bundle structures on topological manifolds homotopy equivalent to S4k−1-bundles over S4k
BibTeX
@misc{lee2026smooth,
title = {{Smooth and Topological Nonorientable Four-Genera of Low-Crossing Knots}},
author = {Seong-Jin Lee},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04243},
eprint = {2611.04243},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04243},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Smooth and Topological Nonorientable Four-Genera of Low-Crossing Knots
AU - Seong-Jin Lee
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04243
UR - https://arxiv.org/abs/2611.04243
AB - We compare linking-form bounds with the smooth and locally flat nonorientable 4-genera of low-crossing knots. For every prime knot with at most 13 crossings whose Witt-level linking-form bound is at least 2, this bound equals both genera. We also exhibit 41 prime knots with 12 or 13 crossings that bound locally flat Möbius bands but no smooth ones, and whose smooth nonorientable 4-genus is 2. Our methods combine linking forms, Heegaard Floer correction terms and definite-lattice obstructions with explicit band constructions. They determine 3,300 of the 3,525 values left unknown in KnotInfo on 2 November 2026; the remaining 225 knots have range [1,2]. We obtain new lower bounds of 2 for 2,102 knots with 14 crossings and correct previously reported values for three knots with 11 crossings.
ER -