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 surface $F \subset S<sup>{3}$. We establish the lower bound [ c(K;F) \ge 2\bigl(t(K)-δ(F)\bigr)+1, ] where $t(K)$ is the tunnel number of $K$ and $δ(F)$ is the Heegaard deficiency of $F$. In particular, for any fixed closed surface $F$, the surface crossing number $c(K;F)$ is unbounded over all knots $K$. Furthermore, we construct a family of knots $K_m$ demonstrating that $c(K_m;F) = Θ(t(K_m))$, which shows that this lower bound is asymptotically sharp.

• The paper proves that the surface crossing number satisfies $c(K;F) \ge 2(t(K)-\delta(F))+1$, linking diagrammatic complexity to tunnel number and Heegaard deficiency.
• The proof uses the chain from ascending number to surface bridge number and Heegaard splittings, with Schultens' amalgamation theorem transferring surface data into a bridge decomposition.
• The bound is asymptotically sharp for iterated connected sums of tunnel-number-one knots, while crossing, ascending, and bridge numbers remain unbounded for every fixed closed surface.

The paper establishes a quantitative link between the diagrammatic complexity of a knot on an arbitrary closed surface and the Heegaard-theoretic complexity of its exterior. The central object is the surface crossing number $c(K;F)$, the minimal number of crossings among all regular diagrams of a knot $K$ obtained by isotoping it into a regular neighborhood of a closed surface $F \subset S^3$. The main result is the inequality

$c(K;F) \ge 2\bigl(t(K) - \delta(F)\bigr) + 1,$

where $t(K)$ is the tunnel number and $\delta(F) = g(M_1) + g(M_2) - g(F)$ is the Heegaard deficiency of $F$, for the decomposition $S^3 = M_1 \cup_F M_2$. The bound is shown to be asymptotically sharp via an explicit family of iterated connected sums.

Definitions and setup

A diagram of $K$ on $F$ is obtained by isotoping $K$0 into $K$1 and projecting onto $K$2; the projection must be an immersion with only transverse double points, and crossing data is read from the $K$3 coordinate. When $K$4, the invariant reduces to the classical crossing number. The author emphasizes that computing $K$5 requires optimizing over all ambient isotopies of $K$6 into $K$7: the global position of the knot relative to the essential curves of $K$8 and the topology of the components of $K$9 matters, not merely the local combinatorics of the projection.

Two auxiliary invariants mediate the proof. The surface bridge number $F \subset S^3$0 is defined both diagrammatically (minimal number of over-bridges and under-bridges) and geometrically via the height function $F \subset S^3$1, recovering Doll's generalized bridge number. The surface ascending number $F \subset S^3$2, extending the author's earlier planar invariant, counts the minimal number of crossings encountered first as under-crossings when traversing an oriented diagram from a base point.

Two structural facts frame the analysis. First, the surface crossing number is monotone under compression: if $F \subset S^3$3 is obtained from $F \subset S^3$4 by compression, then $F \subset S^3$5, since a diagram on $F \subset S^3$6 avoids the attaching disks of any 1-handle by general position. Consequently $F \subset S^3$7 for every closed $F \subset S^3$8, and analogous bounds hold for $F \subset S^3$9 and $c(K;F) \ge 2\bigl(t(K) - \delta(F)\bigr) + 1,$0. Second, the Heegaard deficiency satisfies $c(K;F) \ge 2\bigl(t(K) - \delta(F)\bigr) + 1,$1, with $c(K;F) \ge 2\bigl(t(K) - \delta(F)\bigr) + 1,$2 exactly when $c(K;F) \ge 2\bigl(t(K) - \delta(F)\bigr) + 1,$3 is a Heegaard surface of $c(K;F) \ge 2\bigl(t(K) - \delta(F)\bigr) + 1,$4; in general $c(K;F) \ge 2\bigl(t(K) - \delta(F)\bigr) + 1,$5 measures how far $c(K;F) \ge 2\bigl(t(K) - \delta(F)\bigr) + 1,$6 deviates from being a Heegaard surface.

The fundamental inequality

The main theorem is proved via the chain

$c(K;F) \ge 2\bigl(t(K) - \delta(F)\bigr) + 1,$7

The first inequality is a counting argument: choosing a crossing and two base point/orientation data so that it is first met as an over-crossing, the remaining $c(K;F) \ge 2\bigl(t(K) - \delta(F)\bigr) + 1,$8 crossings split between ascending points of the two resulting diagrams, giving $c(K;F) \ge 2\bigl(t(K) - \delta(F)\bigr) + 1,$9, and flipping an ascending point is a local isotopy inside $t(K)$0. The second inequality constructs a $t(K)$1-bridge presentation: the descending diagram embeds in $t(K)$2 with monotonically decreasing height (a 1-bridge presentation of the unknot), and restoring the over/under data at the $t(K)$3 ascending points introduces exactly one maximum each.

The third and deepest step uses Heegaard theory. Minimal Heegaard splittings $t(K)$4 of the pieces $t(K)$5 are amalgamated via Schultens' theorem, producing a Heegaard surface $t(K)$6 of $t(K)$7 with genus exactly $t(K)$8. Since $t(K)$9 is 1-dimensional and the 1-handles have 2-dimensional attaching disks, general position places $\delta(F) = g(M_1) + g(M_2) - g(F)$0 disjoint from the tubing, so $\delta(F) = g(M_1) + g(M_2) - g(F)$1 admits a $\delta(F) = g(M_1) + g(M_2) - g(F)$2-decomposition with respect to $\delta(F) = g(M_1) + g(M_2) - g(F)$3 with $\delta(F) = g(M_1) + g(M_2) - g(F)$4 and $\delta(F) = g(M_1) + g(M_2) - g(F)$5. The Morimoto–Sakuma–Yokota bound $\delta(F) = g(M_1) + g(M_2) - g(F)$6 then yields $\delta(F) = g(M_1) + g(M_2) - g(F)$7.

An immediate consequence is a realization theorem: for every closed surface $\delta(F) = g(M_1) + g(M_2) - g(F)$8 and every $\delta(F) = g(M_1) + g(M_2) - g(F)$9, some knot satisfies $F$0, so $F$1 for every fixed $F$2. The same chain implies $F$3. Notably, the geometric obstruction in the chain is located in $F$4: the crossing and ascending numbers are diagrammatic, but their unboundedness is driven by the genuinely three-dimensional bridge complexity.

Asymptotic sharpness

The lower bound is asymptotically tight. For $F$5, the $F$6-fold connected sum of a tunnel number one knot such as the trefoil, the results of Morimoto and Scharlemann–Schultens give $F$7, so

$F$8

Conversely, monotonicity under compression and subadditivity of the planar crossing number give $F$9. Hence $S^3 = M_1 \cup_F M_2$0: the surface crossing number grows linearly in the tunnel number, and no lower bound of higher order can hold in general. This linear behavior is not universal — torus knots and 2-bridge knots have unbounded planar crossing number but tunnel number one — so $S^3 = M_1 \cup_F M_2$1 is not a function of $S^3 = M_1 \cup_F M_2$2 alone for arbitrary knots.

Comparisons and extensions

The paper contrasts $S^3 = M_1 \cup_F M_2$3 and $S^3 = M_1 \cup_F M_2$4 with the surface trunk $S^3 = M_1 \cup_F M_2$5, which satisfies $S^3 = M_1 \cup_F M_2$6 but is bounded below by no linear function of $S^3 = M_1 \cup_F M_2$7: by Davies–Zupan, $S^3 = M_1 \cup_F M_2$8, so the trunk stays constant along the family $S^3 = M_1 \cup_F M_2$9 while the tunnel number diverges. The crossing and bridge numbers thus detect global 3-manifold constraints that the trunk does not.

Two generalizations are sketched. For compact orientable surfaces with boundary, replacing handlebodies by compression bodies yields $K$0, where $K$1 is the exterior of $K$2. For spatial graphs, the general position argument in the amalgamation step depends only on 1-dimensionality, and an analogous bound with a graph-dependent additive constant is expected.

The manifold hypothesis is shown to be essential. Dynnikov's 3-page books and Ghrist's universal branched surface both contain every knot with zero crossing number, so singular 2-complexes evade the obstruction entirely; only genuine unbranched surfaces force diagrammatic complexity to reflect the exterior's Heegaard complexity.

Limitations and open questions

Several points are left open. Additivity of $K$3 and $K$4 under connected sum is undetermined for general $K$5, as is additivity of $K$6 when $K$7 is not a Heegaard surface (Doll's additivity covers only the Heegaard case). The linear relation $K$8 is established only for the connected sum family, and the paper asks which classes of prime knots satisfy it, and under what conditions $K$9 is linearly bounded above by $F$0. For surfaces of positive genus, $F$1 alone cannot bound the triangulation complexity of the exterior — a crossing-free diagram can wind through the handles of $F$2 — motivating a refined "compressing diagrammatic complexity" $F$3 and the question of whether it linearly bounds the Haken number. Finally, the extension of the Kauffman–Murasugi–Thistlethwaite theorem to closed surfaces is posed as a conjecture: whether a reduced alternating diagram with cellular embedding on $F$4 realizes $F$5.

Conclusion

This paper proves that the surface crossing number of a knot on any closed surface in $F$6 dominates a linear function of the tunnel number penalized by the Heegaard deficiency of the surface, via a chain running through the ascending and bridge numbers and Schultens' amalgamation theorem. The bound is asymptotically sharp for iterated connected sums, establishing that surface diagrammatic complexity grows linearly with exterior complexity at best, while the realization theorem shows it is unbounded for every fixed surface. The work positions the surface bridge number as the geometric locus of the obstruction and leaves the additivity, prime-knot asymptotics, Haken-type bounds, and alternating minimality on surfaces as concrete open problems.