[Paper Review] Relative Tutte Polynomials for Colored Graphs and Virtual Knot Theory

Yuanan Diao, Gábor Hetyei|ArXiv.org|Sep 7, 2009

Geometric and Algebraic Topology15 references4 citations

TL;DR

This paper introduces the relative Tutte polynomial for colored graphs, generalizing Tutte's original polynomial to allow independent edge operations on a subset of edges. It establishes that the Kauffman bracket polynomial—hence the Jones polynomial—of a virtual knot can be computed directly from the relative Tutte polynomial of its face graph via specific variable substitutions, offering a new, direct method alternative to ribbon graph approaches in virtual knot theory.

ABSTRACT

We introduce the concept of a relative Tutte polynomial of colored graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial to virtual knot theory. More specifically, we show that the Kauffman bracket polynomial (hence the Jones polynomial) of a virtual knot can be computed from the relative Tutte polynomial of its face (Tait) graph with some suitable variable substitutions. Our method offers an alternative to the ribbon graph approach, using the face graph obtained from the virtual link diagram directly.

Motivation & Objective

• To generalize the classical Tutte polynomial to a relative version that treats a subset of edges (H) differently from the rest, preserving labeling independence.
• To provide a labeling-independent relative Tutte polynomial for colored graphs, extending the framework of Bollobás and Riordan to a relative setting.
• To establish a direct connection between the relative Tutte polynomial of a virtual link's face graph and its Kauffman bracket polynomial.
• To offer an alternative to the ribbon graph approach in virtual knot theory by using the face graph directly.
• To enable efficient computation of Jones polynomials for virtual knots via the relative Tutte polynomial, especially for knots formed by tangle replacements.

Proposed method

• Define the relative Tutte polynomial $ T_{/mathcal{H}}(G) $ for a colored graph $ G $ with a subset $ \mathcal{H} $ of edges, using a two-stage process: contracting/deleting edges not in $ \mathcal{H} $, then assigning variables to the remaining graphs via a distinct rule.
• Adopt a spanning tree expansion approach similar to Tutte’s original method, counting edge activities with respect to a labeling order, ensuring independence from the labeling choice.
• Use variable substitutions $ X_+, Y_+, x_+, y_+ $ corresponding to edge types (bridge, loop, etc.) in the face graph, with values $ X_+ = -A^{-3}, Y_+ = -A^3, x_+ = A, y_+ = A^{-1} $, to map to the Kauffman bracket.
• Apply the relative Tutte polynomial to the face (Tait) graph of a virtual link diagram, showing that $ \langle K \rangle = T_{\mathcal{H}}(G) $ under these substitutions.
• Prove the invariance of the relative Tutte polynomial under edge ordering via induction on the number of non-\mathcal{H} edges, using the standard Tutte recurrence for edge deletion and contraction.
• Demonstrate the method on a virtual knot with two virtual crossings, computing the relative Tutte polynomial and verifying it matches the known Kauffman bracket and Jones polynomial.

Experimental results

Research questions

• RQ1 Can a relative Tutte polynomial be defined for colored graphs such that it remains independent of edge labeling, even when only a subset of edges is subject to contraction/deletion?
• RQ2 How can the relative Tutte polynomial of a face graph of a virtual link be used to compute the Kauffman bracket polynomial of the corresponding virtual knot?
• RQ3 Does the relative Tutte polynomial provide a direct, face-graph-based alternative to the ribbon graph method for computing Jones polynomials of virtual knots?
• RQ4 Can the relative Tutte polynomial be used to efficiently compute Jones polynomials for virtual knots constructed via repeated tangle replacement?
• RQ5 What is the precise set of relations that ensure labeling independence in the relative Tutte polynomial, generalizing the Bollobás-Riordan framework?

Key findings

• The relative Tutte polynomial $ T_{\mathcal{H}}(G) $ is independent of the edge labeling order, generalizing Tutte’s original labeling-invariance property to a relative setting.
• The Kauffman bracket polynomial of a virtual knot $ K $ satisfies $ \langle K \rangle = T_{\mathcal{H}}(G) $ under the variable substitutions $ X_+ = -A^{-3}, Y_+ = -A^3, x_+ = A, y_+ = A^{-1} $, where $ G $ is the face graph of $ K $.
• For a virtual knot with writhe 3 and face graph $ G $, the Jones polynomial is computed as $ J_K(t) = t + t^3 - t^4 $, derived from $ \langle K \rangle = -A^{-3} + A^{-7} - A^5 $ via $ A = t^{-1/4} $.
• The method provides a direct computation of the Jones polynomial from the face graph without requiring the construction of a ribbon graph, offering a simpler and more intuitive approach.
• The relative Tutte polynomial generalizes the set-pointed Tutte polynomial and subsumes previous generalizations, making it the most general labeling-independent Tutte polynomial for colored graphs.
• The framework enables polynomial-time computation of Jones polynomials for virtual knots formed by repeated tangle replacement, extending results from classical knot theory to the virtual setting.