[Paper Review] The Kauffman bracket and the Bollobas-Riordan polynomial of ribbon graphs

Sergei Chmutov, Igor Pak|arXiv (Cornell University)|Apr 27, 2004

Geometric and Algebraic Topology12 references7 citations

TL;DR

This paper establishes a direct algebraic relationship between the Kauffman bracket of links in the thickened surface $G \times I$ and the Bollobás-Riordan polynomial of ribbon graphs. By constructing a medial ribbon graph from a link diagram, the authors prove that the Kauffman bracket is an evaluation of the Bollobás-Riordan polynomial under specific variable substitutions, generalizing the classical Tutte polynomial relation to surfaces and extending Kauffman's program to higher genus settings.

ABSTRACT

For a ribbon graph $G$ we consider an alternating link $L_G$ in the 3-manifold $G imes I$ represented as the product of the oriented surface $G$ and the unit interval $I$. We show that the Kauffman bracket $[L_G]$ is an evaluation of the recently introduced Bollobas-Riordan polynomial $R_G$. This results generalizes the celebrated relation between Kauffman bracket and Tutte polynomial of planar graphs.

Motivation & Objective

• To extend Kauffman's program from planar graphs to ribbon graphs on surfaces.
• To establish a precise algebraic relationship between the Kauffman bracket of links in $G \times I$ and the Bollobás-Riordan polynomial of ribbon graphs.
• To generalize the classical relation between the Jones polynomial and the Tutte polynomial to non-planar, surface-embedded links.
• To provide a topological and combinatorial framework for understanding link invariants in 3-manifolds via graph polynomials on surfaces.

Proposed method

• Construct a medial ribbon graph $G$ from a link diagram $\widetilde{L}$ in $G \times I$, where crossings correspond to edges.
• Define a one-to-one correspondence $\varphi$ between states $S$ of the Kauffman bracket and spanning subgraphs $F \subseteq G$, mapping $A$-splittings to edge inclusion and $B$-splittings to edge deletion.
• Express the Kauffman bracket $[\widetilde{L}]$ as a sum over states, with terms $A^{\alpha(S)} B^{\beta(S)} d^{\delta(S)-1}$, where $\delta(S)$ is the number of components after splitting.
• Relate the parameters $\alpha(S)$, $\beta(S)$, and $\delta(S)$ to graph invariants: $\alpha(S) = e(F)$, $\beta(S) = e(G) - e(F)$, $\delta(S) = \mathrm{bc}(F)$.
• Substitute variables in the Bollobás-Riordan polynomial $R_G(x,y,z)$ via $x = Bd/A$, $y = Ad/B$, $z = 1/d$, and scale by $A^{r(G)}B^{n(G)}d^{k(G)-1}$ to match the Kauffman bracket terms.
• Verify that the resulting expression matches the Kauffman bracket term $A^{e(F)}B^{e(G)-e(F)}d^{\mathrm{bc}(F)-1}$, proving the main theorem.

Experimental results

Research questions

• RQ1 How can the Kauffman bracket of a link in $G \times I$ be expressed as a polynomial invariant of the underlying ribbon graph $G$?
• RQ2 What is the precise algebraic relationship between the Bollobás-Riordan polynomial of a ribbon graph and the Kauffman bracket of its associated link?
• RQ3 Does the classical relation between the Jones polynomial and the Tutte polynomial extend to links in thickened surfaces via the Bollobás-Riordan polynomial?
• RQ4 Can the Bollobás-Riordan polynomial serve as a unifying invariant for link invariants in 3-manifolds fibered over the circle?

Key findings

• The Kauffman bracket $[\widetilde{L}_G]$ of the link $L_G$ in $G \times I$ is equal to the evaluation of the Bollobás-Riordan polynomial $R_G$ under the substitutions $x = Bd/A$, $y = Ad/B$, $z = 1/d$, scaled by $A^{r(G)}B^{n(G)}d^{k(G)-1}$.
• The correspondence $\varphi: \mathcal{S}(\widetilde{L}_G) \to \mathcal{F}(G)$ maps states to spanning subgraphs, with $\alpha(S) = e(F)$, $\beta(S) = e(G) - e(F)$, and $\delta(S) = \mathrm{bc}(F)$.
• The resulting expression after substitution matches exactly the term $A^{e(F)}B^{e(G)-e(F)}d^{\mathrm{bc}(F)-1}$ in the Kauffman bracket sum.
• The proof confirms that the Bollobás-Riordan polynomial generalizes the Tutte polynomial in the context of surface-embedded graphs and link invariants.
• The result extends Kauffman’s original planar relation to arbitrary oriented surfaces, providing a topological interpretation of the Bollobás-Riordan polynomial as a link invariant.
• The framework supports generalizations to signed ribbon graphs and suggests potential extensions to colored ribbon graphs and HOMFLY-type invariants.