Crossing Numbers of Knots on Closed Surfaces
Abstract
Let c(K;F) denote the surface crossing number of a knot K with respect to a closed connected surface F in S^3. We relate c(K;F) to the tunnel number t(K) and to the Heegaard deficiency delta(F)=g(M_1;F)+g(M_2;F)-g(F), where S^3=M_1 union_F M_2. The zero-crossing case gives a structural obstruction: if c(K;F)=0, then t(K) <= delta(F). Conversely, if t(K)>delta(F), then c(K;F) >= 2(t(K)-delta(F))+1. Thus the Heegaard deficiency of F measures the amount of tunnel complexity that can be absorbed by F without producing crossings. The proof combines a surface ascending-number estimate, a bridge-number estimate for surface diagrams, and an amalgamation argument for Heegaard splittings relative to F. We also construct connected-sum families showing that the lower bound has the correct linear order.
Cite
@article{arxiv.2602.21659,
title = {Crossing Numbers of Knots on Closed Surfaces},
author = {Makoto Ozawa},
journal= {arXiv preprint arXiv:2602.21659},
year = {2026}
}
Comments
17 pages, 1 figure. Revised version: corrected the zero-crossing case and strengthened the proofs of the surface bridge and amalgamation lemmas