On the crossing number of knots and links on surface in 3-manifolds

Abstract: We give a lower bound of the crossing number of knots and links projected on a 2-sided surface in a 3-manifold using the rank of fundamental groups.

• The paper introduces a novel inequality linking the rank of a link’s exterior fundamental group to diagrammatic parameters, thereby establishing lower bounds on crossing numbers.
• It applies combinatorial and group-theoretic techniques, including generating numbers and HNN extensions, to analyze knots, links, and spatial graphs projected on surfaces.
• The results imply that for any two-sided surface in a compact 3-manifold, crossing numbers can be made arbitrarily high, unifying previous findings and opening new research directions.

Crossing Number Bounds for Knots and Links Projected on Surfaces in 3-Manifolds

Overview

This paper addresses the crossing number of knots and links when projected onto a two-sided surface embedded in a compact 3-manifold. It establishes new lower bounds on the crossing number using combinatorial diagrammatic properties and the ranks of fundamental groups, generalizing prior results and providing a unified framework for understanding diagrammatic complexity in terms of the fundamental group structure. The analysis encompasses knots, links, and spatial graphs within arbitrary compact 3-manifolds, considering both orientable and non-orientable cases as well as manifolds with boundaries.

Main Results and Claims

The core contribution is an inequality that relates the rank of the link group $G(L)$—the fundamental group of the link exterior—with diagrammatic parameters and topological invariants of the ambient manifold and surface. The central theorem states:

$r(G(L)) \leq r(F \subset M) + x(D) - 1$

where $x(D)$ is the generating number of the diagram $D$, and $r(F \subset M)$ is an invariant defined in terms of the ranks of fundamental groups of the components of $M \setminus F$. This result leads directly to a lower bound on the crossing number of the link projected onto the surface:

$$c(L; F \subset M) \geq 2(r(G(L)) - r(F \subset M) - \# D)$$

with $\# D$ the number of diagram components as a subspace of $F$.

A bold claim is that for any two-sided surface $F$ in any compact 3-manifold $r(G(L)) \leq r(F \subset M) + x(D) - 1$0, the $r(G(L)) \leq r(F \subset M) + x(D) - 1$1-crossing number is unbounded. Specifically, for every $r(G(L)) \leq r(F \subset M) + x(D) - 1$2, there exists a projectible knot $r(G(L)) \leq r(F \subset M) + x(D) - 1$3 such that $r(G(L)) \leq r(F \subset M) + x(D) - 1$4. This extends Ozawa's prior results (“Crossing Numbers of Knots on Closed Surfaces” (Ozawa, 25 Feb 2026)), which had established similar bounds for knots in $r(G(L)) \leq r(F \subset M) + x(D) - 1$5.

The paper further generalizes these results to spatial graphs, showing that analogous inequalities hold, with diagrammatic and group-theoretic parameters appropriately adjusted.

Technical Approach

The methodology leverages fundamental group ranks as proxies for complexity. For a link $r(G(L)) \leq r(F \subset M) + x(D) - 1$6 projectible on $r(G(L)) \leq r(F \subset M) + x(D) - 1$7, the paper defines the generating number $r(G(L)) \leq r(F \subset M) + x(D) - 1$8 of an $r(G(L)) \leq r(F \subset M) + x(D) - 1$9-diagram via a minimal set of “generating” regions whose recursive derivation via local crossing rules covers all regions. Using combinatorial arguments and the manner in which regions are glued in the ambient manifold, the author constructs iterated HNN extensions or amalgamated free products (depending on whether $x(D)$0 is separating) for the fundamental group of the relevant manifold pieces.

Key steps include:

• Precise definition and bounding of the generating number $x(D)$1 via inductive corner-counting arguments.
• Lemma relating the number of regions $x(D)$2 to the number of crossings $x(D)$3 and diagram components.
• Algebraic group presentations generalizing Dehn’s approach and using relations induced by local diagrammatic structure.
• Generalization to spatial graphs; diagram crossings and graph vertices are carefully distinguished, and corresponding bounds and group-theoretic constructions are provided.

The bounds are shown to subsume classical results, e.g., the relationship between the bridge index and knot group rank, by specializing the diagram and surface.

Numerical Implications

The inequalities explicitly highlight the role of topological complexity (rank of $x(D)$4) in enforcing crossing number lower bounds when projecting links onto surfaces. The results imply that for links whose fundamental group exteriors have arbitrarily large rank, the crossing number can be forced arbitrarily high, regardless of the surface choice.

For spatial graphs, the lower bound is similarly explicit:

$x(D)$5

with $x(D)$6 the first Betti number of the underlying graph.

Practical and Theoretical Implications

Practically, these results provide tools for certifying diagrammatic complexity and for algorithmic detection of minimal crossing numbers in generalized 3-manifold settings. The bounds are applicable to any compact 3-manifold and any properly embedded two-sided surface, making them relevant for computational knot theory, low-dimensional topology, and applications in DNA topology or fluid dynamics where links are considered in generalized spaces.

Theoretically, the unboundedness result refines understanding of the limitations of surface projections for representing link complexity. It clarifies the interplay between geometric topology and combinatorial knot theory, showing that algebraic invariants constrain diagrammatic minimality in a robust way. The results also suggest pathways for group-theoretic analysis of minimal diagrams for links in branched surfaces or beyond, connecting to universal templates and constructions as in Ghrist [Gh].

Open questions remain regarding optimization of generating sets, tightness of bounds in specific manifolds and surfaces, and possible extension to more general branched surfaces or structures supporting links.

Future Directions

The framework established here invites further investigation into diagrammatic optimization in new 3-manifold settings and more refined algebraic invariants. Extensions to surfaces with more complicated topology, links with prescribed group properties, and connections with the Heegaard genus and tunnel numbers may yield sharper bounds. Algorithmic approaches for computing generating numbers and regions in large diagrams could provide practical improvements for knot tabulation and complexity certification.

Further, the adaptation to spatial graphs suggests a rich interplay between graph-theoretic invariants and 3-manifold topology, which may be relevant for understanding embeddings and decompositions, or for quantum topological invariants.

Conclusion

The paper rigorously establishes group-theoretic lower bounds for the crossing number of knots, links, and spatial graphs projected onto surfaces in compact 3-manifolds. The results generalize and unify prior combinatorial and diagrammatic inequalities, demonstrating unboundedness and providing explicit constraints in terms of fundamental group ranks and diagram structure. This advances the understanding of the complexity of surface projections and their algebraic constraints, with implications for both theoretical topology and algorithmic knot theory.