Duke Mathematical Journal

Alternating links and definite surfaces

Joshua Evan Greene

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We establish a characterization of alternating links in terms of definite spanning surfaces. We apply it to obtain a new proof of Tait’s conjecture that reduced alternating diagrams of the same link have the same crossing number and writhe. We also deduce a result of Banks and of Hirasawa and Sakuma about Seifert surfaces for special alternating links. The appendix, written by Juhász and Lackenby, applies the characterization to derive an exponential time algorithm for alternating knot recognition.

Article information

Duke Math. J., Volume 166, Number 11 (2017), 2133-2151.

Received: 26 January 2016
Revised: 21 October 2016
First available in Project Euclid: 5 May 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 11H55: Quadratic forms (reduction theory, extreme forms, etc.) 05C21: Flows in graphs 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 57M15: Relations with graph theory [See also 05Cxx] 57M27: Invariants of knots and 3-manifolds

alternating links definite surfaces


Greene, Joshua Evan. Alternating links and definite surfaces. Duke Math. J. 166 (2017), no. 11, 2133--2151. doi:10.1215/00127094-2017-0004. https://projecteuclid.org/euclid.dmj/1493971214

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