# Invariants in Low-Dimensional Topology

### Louis H. Kauffman

University of Illinois at Chicago, United States### Simon A. King

Friedrich-Schiller-Universität Jena, Germany### Vassily O. Manturov

Bauman Moscow State Technical University, Russian Federation### Józef H. Przytycki

George Washington University, United States

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## Abstract

The workshop \emph{Invariants in low-dimensional topology}, organised by Louis H. Kauffman (Chicago), Simon A. King (Jena), Vassily O. Manturov (Moscow) and Jozef Przytycki (Washington) was held May 4th--May 10th, 2008. This meeting was attended by 46 participants, including various researchers with recent PhD, one PhD student and one graduate student. The main objective of this workshop was to look intensively at the present state of the art of invariants in low dimensional topolgy, particularly invariants of knots and three dimensional manifolds. This field is rapidly growing with a remarkable influx of new ideas and techniques. This activity was originally inspired in the 1980's by the discovery of the Jones polynomial. Vaughan Jones formulated his invariant in terms of braids and von Neumann algebras and also gave a formula for it via a skein relation (a relation among knot and link diagrams) that was a clear generalization of the Conway skein relation for the classical Alexander polynomial. This led at once to the discovery by many people of a generalization called the Homflypt polynomial, and then sometime later another skein theoretic generalization --- the Kauffman polynomial --- and to a general skein theory for knots in three-manifolds (skein modules and skein algebras) due to Jozef Przytycki. The Jones polynomial is related to statistical mechanics. A key notion from statistical mechanics that was imported here was the idea to write an invariant in analogy to a partition function. This means that the invariant is expressed as a sum over combinatorial states related either to a link diagram or a triangulation of a three-manifold. The first example of this is the state sum for the Jones polynomial called Kauffman bracket. The state sums generalize to sums involving solutions to the Yang-Baxter Equation (originally used in statistical mechanics) and this in turn led to constructions of invariants from quantum groups and Hopf algebras. Then in the latter part of the 1980's ideas from quantum field theory (and the work of Witten) crystallized in the notion of Topological Quantum Field Theories (due to Witten and Atiyah) and corresponding invariants of knots and three-manifolds. At about the same time Vassiliev and Goussarov introduced new ideas about the space of knots and associated invariants that were quickly reformulated through the work of Birman, Lin, Kontsevich and Bar-Natan yielding deep relationships with Lie algebras and new ways to understand the previous constructions involving quantum groups. In 1996 Kauffman introduced a generalization of knot theory called virtual knot theory, which is given diagrammatically, and is equivalent to a stabilized theory of knots in thickened surfaces. At the same time there was an evolution in the concept of the fundamental group of the knot complement with Joyce's introduction of the quandle in 1979 and Matveev's independent discovery of this structure (distributive groupoids). After 2000 Fenn, Kauffman, Manturov and others generalized the quandle to a stronger invariant called the biquandle which is very important for virtual knots. In 1998 Khovanov introduced a categorification of the Jones polynomial. This means that he constructed a homology theory based on link diagrams whose graded Euler characteristic equals the original Jones polynomial. This led eventually to an explosion of significant results and new constructions. Khovanov and Rozansky found ways to categorify the Homflypt polynomial, and at this conference Rozansky announced the categorification of the Kauffman 2-variable polynomial. After Khovanov's original discovery, a categorification of the Alexander polynonmial (Knot Floer Homology) was found by Ozsvath and Szabo (2002) and eventually made purely combinatorial by Manolescu, Ozsvath and Sarkar. This categorification of the Alexander polynomial has led one of the most startling results in knot theory: The minimal genus of an orientable spanning surface for a knot, can be calculated from the Knot Floer homology, and this is a finite combinatorial matter in terms of a diagram for the knot. This conference collected many active researchers in this field with talks on all of the topics mentioned above. We shall now mention the main themes of the workshop and the participants who were influential or gave talks on these themes: \begin{itemize} \item \textbf{TQFT and skein theory} Marta Asaeda, Christian Blanchet, Charles Frohman, Patrick Gilmer, Uwe Kaiser, Joanna Kania-Bartoszy\'nska, \linebreak Thomas Kerler, Gregor Masbaum, Michael M\"uger, Jozef Przytycki and Adam Sikora.\pagebreak \item \textbf{Knots, braids, and geometric topology} Simon King, Elena Kudryavtseva, Sofia Lambropoulou, S\'ostenes Lins, Sergei Matveev, Cameron McA. Gordon, Wolfgang Metzler, Hugh Morton, Carlo Petronio, Michael Polyak, Nikolai Saveliev, Radmila Sazdanovic, Pawel Traczyk, Vladimir Vershinin and Oleg Viro. \item \textbf{Categorified invariants and link homology} Marta Asaeda, Dror Bar-Natan, Anna Beliakova, Christian Blanchet, Mikhael Chmutov, Karl-Mag\-nus Jacobsson, Vassily Manturov, Jozef Przytycki, Yongwu Rong, Lev Rozansky, Radmila Sazdanovic and Alexander Shumakovitch. \item \textbf{Quandles, Biquandles and Algebraic Categorification} Dror Bar-Natan, Scott Carter, Alissa Crans, Roger Fenn, Sergei Matveev, Maciej Niebrzydowski, Jozef Przytycki and Masahico Saito. \item \textbf{Virtual and Welded Knots} Dror Bar-Natan, Heather Dye, Roger Fenn, Louis Kauffman and Lev Rozansky. \item \textbf{State Sums} Sergei Chmutov, Heather Dye, Louis Kauffman, Simon King, Sostenes Lins, Sergei Matveev and Maciej Mroczkowski. \item \textbf{Hopf and Frobenius Algebras and three-manifold invariants} Uwe Kaiser, Louis Kauffman, Thomas Kerler, Michael M\"uger and David Radford. \item \textbf{Vassiliev Invariants} Dror Bar-Natan, Sergei Chmutov, Sergei Duzhin and Michael Polyak. \end{itemize} In our opinion, many of the results in the conference were new and significant. Having the conference at the Mathematisches Forschungsinstitute enabled the participants not only to listen to new results but also to the opportunity to converse and do mathematics together. \noindent The conference was dedicated to the 60th birthday of Oleg Yannovich Viro.