The Jones polynomial and graphs on surfaces

Abstract

The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the Tutte polynomial of planar graphs to graphs that are embedded in closed oriented surfaces of higher genus. In this paper we show that the Jones polynomial of any link can be obtained from the Bollobas-Riordan-Tutte polynomial of a certain oriented ribbon graph associated to a link projection. We give some applications of this approach.

19 pages, 9 figures, minor changes

References in corpus (4)

• A volume-ish theorem for the Jones polynomial of alternating knots
• The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial
• Minimal diagrams of classical knots
• The Kauffman bracket and the Bollobas-Riordan polynomial of ribbon graphs

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• Unsigned state models for the Jones polynomial
• A characterization of partially dual graphs
• Partial duals of plane graphs, separability and the graphs of knots
• Non-orientable quasi-trees for the Bollobas-Riordan polynomial
• Turaev genus and alternating decompositions
• Expansions for the Bollobas-Riordan polynomial of separable ribbon graphs
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• Towards tangle calculus for Khovanov polynomials
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• Adequate links in thickened surfaces and the generalized Tait conjectures
• Heegaard diagrams corresponding to Turaev surfaces
• The dealternating number and the alternation number of a closed 3-braid
• Extremal Khovanov homology of Turaev genus one links
• Quantum Statistical Mechanics of the Absolute Galois Group
• Unknown values in the table of knots
• The upsilon invariant at 1 of 3-braid knots
• Hopf algebras and Tutte polynomials
• Introduction to Graph-Link Theory
• A reduced set of moves on one-vertex ribbon graphs coming from links
• Volumes of Montesinos links
• Kauffman's clock lattice as a graph of perfect matchings: a formula for its height
• Graphical methods establishing nontriviality of state cycle Khovanov homology classes
• Topological Tutte Polynomial
• Alternating links on surfaces and volume bounds
• The boundaries of dipole graphs and the complete bipartite graphs K_{2,n}
• The Jones polynomials of 3-bridge knots via Chebyshev knots and billiard table diagrams
• Generalized duality for graphs on surfaces and the signed Bollobas-Riordan polynomial
• Excluded minors and the ribbon graphs of knots
• Jones polynomials, volume, and essential knot surfaces: a survey
• A geometric characterization of the upper bound for the span of the Jones polynomial
• The Khovanov width of twisted links and closed 3-braids
• A survey on the Turaev genus of knots
• Almost alternating knots with 12 crossings and Turaev genus
• Cusp Volumes of Alternating Knots on Surfaces
• A remark on a finiteness of purely cosmetic surgeries
• On the homology theory for the chromatic polynomials
• Turaev genus, knot signature, and the knot homology concordance invariants
• Dehn filling, volume, and the Jones polynomial
• Near extremal Khovanov homology of Turaev genus one links
• Span of the Jones polynomials of certain v-adequate virtual links
• Characterizations of Eulerian and even-face partial duals of ribbon graphs
• A note on thickness of knots
• The Jones polynomial of an almost alternating link
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• State Surfaces of Links
• A counterexample to Question 1 of "A survey on the Turaev genus of knots"
• A New Polynomial for Checkerboard-Colorable 4-Valent Virtual Graphs
• On the arc index and Turaev genus of a link
• On the degree of the colored Jones polynomial
• State generatings for Jones and Kauffman-Jones polynomials
• Combinatorics of Link Diagrams and Volume
• The 2-Factor Polynomial Detects Even Perfect Matchings
• Topological quantum field theory and polynomial identities for graphs on the torus
• State graphs and fibered state surfaces
• Counterexamples to conjectures by Gross, Mansour and Tucker on partial-dual genus polynomials of ribbon graphs