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A multi-variable Alexander polynomial for a framed transverse graph

Yuanyuan Bao, Zhongtao Wu

TL;DR

The paper extends the Alexander polynomial to framed, oriented transverse graphs in S^3 by introducing an extended rotation number for graph diagrams and a normalized multi-variable state-sum invariant Δ_𝔾. It establishes topological invariance under framed Reidemeister moves and shows that Δ_𝔾 coincides with Viro's multi-variable U_q(gl(1|1))-Alexander polynomial after a specified variable change, providing a combinatorial interpretation of Viro's invariant. The approach integrates Kauffman state sums, MOY graph theory, and a normalization based on rotation data, yielding a robust framework for transverse graph invariants. This work deepens connections between diagrammatic combinatorics and representation-theoretic quantum invariants, with potential implications for spatial graph theory and low-dimensional topology.

Abstract

We propose a definition of the rotation number for transverse graph diagrams, extending the classical notion of the rotation number for plane curves. Using this, we introduce a normalized multi-variable Alexander polynomial for framed, oriented transverse graphs without sinks or sources, embedded in the 3-sphere $S^3$. We prove that our invariant coincides with the $U_q(\mathfrak{gl}(1\vert 1))$-Alexander polynomial proposed by Viro.

A multi-variable Alexander polynomial for a framed transverse graph

TL;DR

The paper extends the Alexander polynomial to framed, oriented transverse graphs in S^3 by introducing an extended rotation number for graph diagrams and a normalized multi-variable state-sum invariant Δ_𝔾. It establishes topological invariance under framed Reidemeister moves and shows that Δ_𝔾 coincides with Viro's multi-variable U_q(gl(1|1))-Alexander polynomial after a specified variable change, providing a combinatorial interpretation of Viro's invariant. The approach integrates Kauffman state sums, MOY graph theory, and a normalization based on rotation data, yielding a robust framework for transverse graph invariants. This work deepens connections between diagrammatic combinatorics and representation-theoretic quantum invariants, with potential implications for spatial graph theory and low-dimensional topology.

Abstract

We propose a definition of the rotation number for transverse graph diagrams, extending the classical notion of the rotation number for plane curves. Using this, we introduce a normalized multi-variable Alexander polynomial for framed, oriented transverse graphs without sinks or sources, embedded in the 3-sphere . We prove that our invariant coincides with the -Alexander polynomial proposed by Viro.

Paper Structure

This paper contains 17 sections, 22 theorems, 40 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Rotation numbers for transverse graph diagrams
  3. Transverse spatial graphs
  4. Extended rotation number
  5. Classical rotation number for plane curves
  6. Generalization to transverse graphs
  7. Properties of the rotation number
  8. Relations with existing invariants.
  9. A multi-variable Alexander polynomial
  10. Kauffman states
  11. Kauffman state sum
  12. A normalized multi-variable Alexander polynomial
  13. Specialization to MOY graphs
  14. Topological invariance
  15. Relation with Viro's $U_q(\mathfrak{gl}(1\vert 1))$-Alexander polynomial
  16. ...and 2 more sections

Key Result

Theorem 2.2

Two diagrams represent the same transverse graph if and only if they can be transformed into one another by a finite sequence of moves shown in Fig. fig:e25.

Figures (12)

  • Figure 1: The local picture of a vertex with transverse orientation (left). An oriented trivalent graph without sinks or sources is a transverse graph (right).
  • Figure 2: Reidemeister moves for transverse graph diagrams. Suppressed orientations of the edges can be added in all compatible ways.
  • Figure 3: Inductive rule for assigning winding numbers to regions across an edge $e$.
  • Figure 4: Incoming and outgoing edges around a vertex $v$. Gray letters indicate the winding numbers of the regions between consecutive edges.
  • Figure 5: A diagram of a trivalent graph.
  • ...and 7 more figures

Theorems & Definitions (53)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3: Viro Viro1
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Example 2.7
  • Proposition 2.8
  • proof
  • ...and 43 more