## Duke Mathematical Journal

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

### Alternating links and definite surfaces

#### Abstract

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

**Source**

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

**Dates**

Received: 26 January 2016

Revised: 21 October 2016

First available in Project Euclid: 5 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.dmj/1493971214

**Digital Object Identifier**

doi:10.1215/00127094-2017-0004

**Mathematical Reviews number (MathSciNet)**

MR3694566

**Zentralblatt MATH identifier**

1377.57009

**Subjects**

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

**Keywords**

alternating links definite surfaces

#### Citation

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