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.
"Alternating links and definite surfaces." Duke Math. J. 166 (11) 2133 - 2151, 15 August 2017. https://doi.org/10.1215/00127094-2017-0004