Crossing Numbers of Knots on Closed Surfaces

TL;DR

This paper investigates the relationship between the surface crossing number of knots on closed surfaces, tunnel number, and Heegaard deficiency, providing bounds and structural insights into knot complexity on surfaces.

Contribution

It establishes bounds linking surface crossing number, tunnel number, and Heegaard deficiency, and constructs examples confirming the bounds' linear order.

Findings

If c(K;F)=0, then t(K) <= delta(F).

If t(K)>delta(F), then c(K;F) >= 2(t(K)-delta(F))+1.

Constructed families show the lower bound has the correct linear order.

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.