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Invariants of Handlebody-Links and Spatial Graphs

V. G. Bardakov, D. A. Fedoseev

TL;DR

The paper develops a broad algebraic framework linking handlebody-links and spatial graphs to quandle theory by extending standard quandle colorings to generalized systems. It introduces $(X,G,iglrace *_gigrrace,f,igotimes,igoplus)$-systems, a good involution, and trivalent-compatible conditions to produce robust coloring invariants for trivalent graphs and their associated handlebody-links; it also defines the fundamental associated quandle and connects colorings to homomorphisms from this quandle. Key contributions include precise axioms for $(G,*,f)$- and $(Q,∘,f)$-family generalizations, the associative composition requirement for SR-invariance, and Λ-compatible invariants that cover vertices of varying valency. The work clarifies how these invariants behave under the standard moves and provides concrete examples illustrating differences between spatial graphs and handlebody-links, along with open problems for further development. Overall, the framework offers a unified method to derive and study quandle-based invariants across both spatial graphs and handlebody-links with potential Yang–Baxter applications and broader category-theoretic connections.

Abstract

A $G-$family of quandles is an algebraic construction which was proposed by A. Ishii, M. Iwakiri, Y. Jang, K. Oshiro in 2013. The axioms of these algebraic systems were motivated by handlebody-knot theory. In the present work we investigate possible constructions which generalise $G-$family of quandles and other similar constructions (for example, $Q-$ and $(G,*,f)-$families of quandles). We provide the necessary conditions under which the resulting object (called an $(X,G,{*_g},f,\otimes,\oplus)-$system) gives a colouring invariant of knotted handlebodies. We also discuss several other modifications of the proposed construction, providing invariants of spatial graphs with an arbitrary (finite) set of values of vertex valency. Besides, we consider several examples which in particular showcase the differences between spatial trivalent graph and handlebody-link theories.

Invariants of Handlebody-Links and Spatial Graphs

TL;DR

The paper develops a broad algebraic framework linking handlebody-links and spatial graphs to quandle theory by extending standard quandle colorings to generalized systems. It introduces -systems, a good involution, and trivalent-compatible conditions to produce robust coloring invariants for trivalent graphs and their associated handlebody-links; it also defines the fundamental associated quandle and connects colorings to homomorphisms from this quandle. Key contributions include precise axioms for - and -family generalizations, the associative composition requirement for SR-invariance, and Λ-compatible invariants that cover vertices of varying valency. The work clarifies how these invariants behave under the standard moves and provides concrete examples illustrating differences between spatial graphs and handlebody-links, along with open problems for further development. Overall, the framework offers a unified method to derive and study quandle-based invariants across both spatial graphs and handlebody-links with potential Yang–Baxter applications and broader category-theoretic connections.

Abstract

A family of quandles is an algebraic construction which was proposed by A. Ishii, M. Iwakiri, Y. Jang, K. Oshiro in 2013. The axioms of these algebraic systems were motivated by handlebody-knot theory. In the present work we investigate possible constructions which generalise family of quandles and other similar constructions (for example, and families of quandles). We provide the necessary conditions under which the resulting object (called an system) gives a colouring invariant of knotted handlebodies. We also discuss several other modifications of the proposed construction, providing invariants of spatial graphs with an arbitrary (finite) set of values of vertex valency. Besides, we consider several examples which in particular showcase the differences between spatial trivalent graph and handlebody-link theories.

Paper Structure

This paper contains 15 sections, 20 theorems, 106 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Extensions of quandles
  3. $G$-family of quandles and $Q$-family of quandles
  4. $Q$-family of quandles and its generalization
  5. $(G,*,f)$-families of quandles and handlebody-links
  6. Handlebody-links
  7. $(X,G,\{*_g\},f,\otimes,\oplus)$-systems
  8. Fundamental associated quandle
  9. Invariants of spatial graphs and handlebody-links
  10. Groups of spatial graphs
  11. Handlebody-links and spatial graphs
  12. Topological man with watch
  13. The fundamental quandle of a spatial graph
  14. An invariant of spatial graphs coming from $(f,\otimes)$-systems
  15. Open problems

Key Result

Proposition 2.5

Consider a $(G,*,f)$-family of quandles and an arbitrary $g\in G$. Then $*_g$ is idempotent and right-invertible.

Figures (13)

  • Figure 1: Moves on handlebody-link diagrams
  • Figure 2: "Topological man" with linked (a) and unlinked (b) fingers, and its graph representation
  • Figure 3: How to unlink the fingers of the topological man in the category of handlebody-links. First figure depicts the graph-representation of the man with linked fingers, while the last is the representation of the man with unlinked fingers
  • Figure 4: Colouring rules for handlebody-links diagrams by a $G$-family of quandles
  • Figure 5: Colouring rules for handlebody-links diagrams in the general case
  • ...and 8 more figures

Theorems & Definitions (54)

  • Definition 2.1
  • Example 2.3
  • Definition 2.4: BF
  • Proposition 2.5
  • proof
  • Lemma 2.6: BF
  • Example 2.7
  • Proposition 2.8
  • proof
  • Remark 2.9
  • ...and 44 more