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 some space 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 knots coming from Yang–Baxter homology. Here the work of Fenn–Rourke–Sanderson (geometric realization of pre-cubic 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.
Dedicated to Lou Kauffman for his 70th birthday.
I decided to keep the original abstract of the talk omitting only the last sentence “But I believe in Open Talks, that is I hope to discuss and develop above topics in an after-talk discussion over coffee or tea with willing participants”, which applies to a talk but not a paper.
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Notes
- 1.
The early Bronze Age in Greece is divided, as in Crete and the Cyclades, into three phases. The second phase lasted from 2500 to 2200 BC, and was marked by a considerable increase in prosperity. There were palaces at Lerna, Tiryns, and probably elsewhere, in contact with the Second City of Troy. The end of this phase (in the Peloponnese) was brought about by invasion and mass burnings. The invaders are thought to be the first speakers of the Greek language to arrive in Greece.
- 2.
Heliodorus was a surgeon in the 1st century AD, probably from Egypt, and mentioned in the Satires of Juvenal. This Heliodorus wrote several books on medical technique which have survived in fragments and in the works of Oribasius [39]. It is worth to cite Miller: “In the ‘Iatrikon Synagogos,’ a medical treatise of Oribasius of Pergamum (...) Heliodorus, who lived at the time of Trajan (Roman Emperor 98–117 AD), also mentions in his work knots and loops” [39, 57].
- 3.
Hippocrates of Cos (c. 460–375 BC). A commentary on the Hippocratic treatise on Joints was written by Apollonios of Citon (in Cypros), who flourished in Alexandria in the first half of the first century BC. That commentary has obtained a great importance because of an accident in its transmission. A manuscript of it in Florence (Codex Laurentianus) is a Byzantine copy of the ninth century, including surgical illustrations (for example, with reference to reposition methods), which might go back to the time of Apollonios and even Hippocrates. Iconographic tradition of this kind are very rare, because the copying of figures was far more difficult than the writing of the text and was often abandoned [57]. The story of the illustrations to Apollonios’ commentary is described in [58].
- 4.
From [59]: “The purpose of Oribasios Medical Collection is so well explained at the beginning of it that it is best to quote his own words”: Autocrator Iulian, I have completed during our stay in Western Gaul the medical summary which your Divinity had commanded me to prepare and which I have drawn exclusively from the writings of Galen. After having praised it, you commanded me to search for and put together all that is most important in the best medical books and all that contributed to attain the medical purpose. I gladly undertook that work being convinced that such a collection would be very useful. (...) As it would be superfluous and even absurd to quote from the authors who have written in the best manner and then again from those who have not written as careful, I shall take my material exclusively from the best authors without omitting anything which I first obtained from Galen....
- 5.
Otherwise the Codex of Nicetas is the earliest surviving illustrated surgical codex, containing 30 full page images illustrating the commentary of Appolonios of Kition and 63 smaller images scattered through the pages.
- 6.
Cyrus Lawrence Day (Dec. 2, 1900–July 5, 1968) was (in 1967) Professor Emeritus of English of the University of Delaware. A graduate of Harvard, he took an M.A. degree at Columbia and returned to Harvard for his PhD degree. (...) Mr. Day, a yachtsman since his boyhood, is the author, also, of a standard book on sailor’s knots [10,11,12].
- 7.
- 8.
Giorgio Vasari writes in [66]: “[Leonardo da Vinci] spent much time in making a regular design of a series of knots so that the cord may be traced from one end to the other, the whole filling a round space. There is a fine engraving of this most difficult design, and in the middle are the words: Leonardus Vinci Academia.”
- 9.
- 10.
The year 1928 is the year of publication of the Alexander’s paper, however already in 1919 he discusses in a letter to Oswald Veblen, his former Ph.D. adviser, “a genuine and rather jolly invariant” which we call today the determinant of the knot. It is this construction which Alexander extends later to the Alexander polynomial \(\Delta _D(t)\) (determinant is equal to \(\Delta _D(t)\) for \(t=-1\)). In fact the Alexander letter contains more: Alexander constructs the space which we call often today the space of nontrivial Fox \({\mathbb Z}\)-colorings or the first homology of the double branched cover of \(S^3\) along the knot [2].
- 11.
We consider arcs from undercrossing to undercrossing and semi-arcs from crossing to crossing.
- 12.
We can think of “inverse” formally: we introduce the monoid of binary operations on X, Bin(X), with composition given by \(a(*_1*_2)b= (a*_1b)*_2b\) and identity element \(*_0\) given by \(a*_0b=a\), then the inverse means the inverse in the monoid, that is \(*\bar{*}=*_0= \bar{*}*\); see [46].
- 13.
The concept was introduced in 1950 by Eilenberg and Zilber under the name semi-simplicial complex [13].
- 14.
Older names include: the star-triangle relation, the triangle equation, and the factorization equation [23].
- 15.
We should stress that to find link invariants it suffices to use directly oriented second and third Reidemeister moves in addition to both first Reidemeister moves, as we can restrict ourselves to braids and use the Markov theorem. This point of view was used in [65].
- 16.
The conference Knots in Hellas I took place in Delphi, Greece in August of 1998, while the e-print of Khovanov work was put on arXiv in August of 1999. However Mikhail Khovanov had already an idea of Khovanov homology in summer of 1997.
- 17.
To get the classical Jones notation we put \(q=-t^{1/2}\).
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Acknowledgements
I would like to thank Sofia Lambropoulou for organizing for the second time the great Knots in Hellas conference.
I was partially supported by the Simons Collaboration Grant-316446 and CCAS Dean’s Research Chair award.
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Przytycki, J.H. (2019). Knot Theory: From Fox 3-Colorings of Links to Yang–Baxter Homology and Khovanov Homology. In: Adams, C., et al. Knots, Low-Dimensional Topology and Applications. KNOTS16 2016. Springer Proceedings in Mathematics & Statistics, vol 284. Springer, Cham. https://doi.org/10.1007/978-3-030-16031-9_5
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