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The crossing and the arc from the topological viewpoint

Igor Nikonov

TL;DR

The paper reframes knot diagrams on surfaces through a topological lens, treating diagram elements (arcs, semiarcs, regions, crossings) as isotopy/homotopy classes of probes in tangle thickenings. It develops a rich categorical framework of diagram categories and probe spaces, and defines strong/weak equivalences and universal invariants/coinvariants (including h-variants) valued in Rel, capturing how elements persist under Reidemeister moves and homotopies. Central constructions include the universal invariant/coinvariant machinery, the crossing homology class via the multicrossing complex, and colorings by topological analogues of quandles, biquandloids, crossoids, and partial ternary quasigroups, tying together tribracket, biquandle, and crossoid homologies. The results provide a unifying topological account for diagram-labeling invariants and their homotopical refinements, enabling a systematic study of colorings, skein-type structures, and potential skein-module frameworks on surfaces. The work also outlines higher-level constructs (multicrossing complex, crossing class) and speculative directions toward new invariants for tangles and knots on surfaces.

Abstract

The combinatorial approach to knot theory treats knots as diagrams modulo Reidemeister moves. Many constructions of knot invariants (e.g., index polynomials, quandle colorings, etc.) use elements of diagrams such as arcs and crossings by assigning invariant labels to them. The universal invariant labels, which carry the most information, can be thought of as equivalence classes of arcs and crossings modulo the relation, which identifies corresponding elements of diagrams connected by a Reidemeister move. We can call these equivalence classes the arcs and crossings of the knot. In the paper, we give a topological description of sets of these classes as the isotopy classes of probes of diagram elements. In the second part of the paper, we discuss the homotopy classes of diagram elements. We demonstrate that the sets of these classes are fundamental for the algebraic objects that are responsible for coloring diagrams of tangles on a given surface. For arcs, these algebraic objects are quandles; for regions, they are partial ternary quasigroups; for semiarcs, they are biquandloids; and for crossings, they are crossoids. The definitions of the last three algebraic structures are given in the paper. Additionally, we introduce the multicrossing complex of a tangle and define the crossing homology class. In a sense, the multicrossing complex unifies tribracket, biquandle and crossoid homologies; and the tribracket, biquandle and crossoid cycle invariants are actually the result of pairing a tribracket (biquangle, crossoid) cocycle with the crossing homology class.

The crossing and the arc from the topological viewpoint

TL;DR

The paper reframes knot diagrams on surfaces through a topological lens, treating diagram elements (arcs, semiarcs, regions, crossings) as isotopy/homotopy classes of probes in tangle thickenings. It develops a rich categorical framework of diagram categories and probe spaces, and defines strong/weak equivalences and universal invariants/coinvariants (including h-variants) valued in Rel, capturing how elements persist under Reidemeister moves and homotopies. Central constructions include the universal invariant/coinvariant machinery, the crossing homology class via the multicrossing complex, and colorings by topological analogues of quandles, biquandloids, crossoids, and partial ternary quasigroups, tying together tribracket, biquandle, and crossoid homologies. The results provide a unifying topological account for diagram-labeling invariants and their homotopical refinements, enabling a systematic study of colorings, skein-type structures, and potential skein-module frameworks on surfaces. The work also outlines higher-level constructs (multicrossing complex, crossing class) and speculative directions toward new invariants for tangles and knots on surfaces.

Abstract

The combinatorial approach to knot theory treats knots as diagrams modulo Reidemeister moves. Many constructions of knot invariants (e.g., index polynomials, quandle colorings, etc.) use elements of diagrams such as arcs and crossings by assigning invariant labels to them. The universal invariant labels, which carry the most information, can be thought of as equivalence classes of arcs and crossings modulo the relation, which identifies corresponding elements of diagrams connected by a Reidemeister move. We can call these equivalence classes the arcs and crossings of the knot. In the paper, we give a topological description of sets of these classes as the isotopy classes of probes of diagram elements. In the second part of the paper, we discuss the homotopy classes of diagram elements. We demonstrate that the sets of these classes are fundamental for the algebraic objects that are responsible for coloring diagrams of tangles on a given surface. For arcs, these algebraic objects are quandles; for regions, they are partial ternary quasigroups; for semiarcs, they are biquandloids; and for crossings, they are crossoids. The definitions of the last three algebraic structures are given in the paper. Additionally, we introduce the multicrossing complex of a tangle and define the crossing homology class. In a sense, the multicrossing complex unifies tribracket, biquandle and crossoid homologies; and the tribracket, biquandle and crossoid cycle invariants are actually the result of pairing a tribracket (biquangle, crossoid) cocycle with the crossing homology class.

Paper Structure

This paper contains 19 sections, 54 theorems, 67 equations, 42 figures.

Table of Contents

  1. Introduction
  2. Diagram
  3. Tangles and tangle diagrams
  4. The probe spaces
  5. Diagram categories
  6. Diagram elements
  7. The crossing and the arc from the combinatorial viewpoint
  8. Invariants and coinvariants
  9. Universal invariants and coinvariants
  10. Universal invariant
  11. Universal local invariant
  12. Universal coinvariant
  13. h-coinvariants
  14. Arc
  15. Weak equivalence and motion group
  16. ...and 4 more sections

Key Result

Proposition 1

Figures (42)

  • Figure 1: A knot diagram
  • Figure 2: Reidemeister moves
  • Figure 3: Elements of a knot diagram
  • Figure 4: Topological interpretation of diagram elements
  • Figure 5: Codimension $1$ singularities: an ordinary cusp, a simple tangency and a triple point
  • ...and 37 more figures

Theorems & Definitions (154)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Definition 3
  • Definition 4
  • Remark 1
  • Proposition 2
  • Proposition 3
  • proof
  • Definition 5
  • ...and 144 more