Knot theory: from fox 3-colorings of links to Yang–Baxter homology and Khovanov homology

Józef Henryk Przytycki

Abstract

This paper is an extended account of my “Introductory Plenary talk at Knots in Hellas 2016” conference. We start from the short introduction to Knot Theory from the historical perspective, starting from Heraclas text (the first century AD), mentioning R. Llull (1232–1315), A. Kircher (1602–1680), Leibniz idea of Geometria Situs (1679), and J.B. Listing (student of Gauss) work of 1847. We spend somespace on Ralph H. Fox (1913–1973) elementary introduction to diagram colorings(1956). In the second section we describe how Fox work was generalized to distributive colorings (racks and quandles) and eventually in the work of Jones and Turaev to link invariants via Yang–Baxter operators; here the importance of statistical mechanics to topology will be mentioned. Finally we describe recent developments which started with Mikhail Khovanov work on categorification of the Jones polynomial. By analogy to Khovanov homology we build homology of distributive structures (including homology of Fox colorings) and generalize it to homology of Yang–Baxter operators. We speculate, with supporting evidence, on co-cycle invariants of knotscoming from Yang–Baxter homology. Here the work of Fenn–Rourke–Sanderson (geometric realization of precubic sets of link diagrams) and Carter–Kamada–Saito (co-cycle invariants of links) will be discussed and expanded. No deep knowledge of Knot Theory, homological algebra, or statistical mechanics is assumed as we work from basic principles. Because of this, some topics will be only briefly described.
Author Józef Henryk Przytycki (FMPI/IM)
Józef Henryk Przytycki,,
- Institute of Mathematics
Pages115-145
Publication size in sheets1.50
Book Adams Colin C., Gordon Cameron McA., Jones Vaughan F. R., Kauffman Louis H., Lambropoulou Sofia, Millett Kenneth C., Przytycki Józef Henryk, Ricca Renzo, Sazdanovic Radmila (eds.): Knots, low-dimensional topology and applications: Knots in Hellas, International Olympic Academy, Greece, July 2016, Springer Proceedings in Mathematics & Statistics, no. 284, 2019, Springer, ISBN 978-3-030-16030-2, [978-3-030-16031-9], 488 p., DOI:10.1007/978-3-030-16031-9
Keywords in Englishknot theory, history of knots, fox colorings, cocycle invariants, Yang-Baxter operator, Khovanov homology, categorification
DOIDOI:10.1007/978-3-030-16031-9_5
URL https://link.springer.com/content/pdf/10.1007%2F978-3-030-16031-9_5.pdf
Languageen angielski
Score (nominal)20
Score sourcepublisherList
ScoreMinisterial score = 20.0, 28-01-2020, ChapterFromConference
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