Abstract
The Turaev genus of a knot is a topological measure of how far a given knot is from being alternating. Recent work by several authors has focused attention on this interesting invariant. We discuss how the Turaev genus is related to other knot invariants, including the Jones polynomial, knot homology theories, and ribbon-graph polynomial invariants.
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Notes
[Added in press] Cody Armond and Moshe Cohen recently informed us that \(q(\mathbb {G};t)\) is different for two diagrams of the knot 821. Question 1 remains open for adequate knots.
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Acknowledgments
We would like to thank the organizers of the Quantum Topology and Hyperbolic Geometry Conference (Nha Trang, Vietnam, May 13–17, 2013) for their extraordinary hospitality. We gratefully acknowledge support by the Simons Foundation and PSC-CUNY.
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Champanerkar, A., Kofman, I. A Survey on the Turaev Genus of Knots. Acta Math Vietnam 39, 497–514 (2014). https://doi.org/10.1007/s40306-014-0083-y
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DOI: https://doi.org/10.1007/s40306-014-0083-y