Students

• Peter Lee's home page.
• Zsuzsanna Dancso's home page.

JforAlternatingTangles.pdf. Source: JforAlternatingTangles.zip	The abstract of my student Hernando Burgos' first paper reads as follows: It is a well known result from Thistlethwaite that the Jones polynomial of a non-split alternating link is alternating. We find the "right" generalization of this result to the case of non-split alternating tangles. More specifically: the Jones polynomial of tangles is valued in a certain skein module, we describe an alternating condition on elements of this skein module, show that it is satisfied by the Jones invariant of the tangles consisting of a single overcrossing or a single undercrossing, and prove that it is preserved by appropriately "alternating" planar algebra compositions. Hence, this condition is satisfied by the Jones polynomial of all alternating tangles. Finally, in the case of 0-tangles, that is links, our condition is equivalent to simple alternation of the coefficients of the Jones polynomial.
arXiv:0711.4001	Based on his master's project, my student Fionntan Roukema wrote an article titled "Goussarov-Polyak-Viro combinatorial formulas for finite type invariants". His abstract reads: Goussarov, Polyak, and Viro proved that finite type invariants of knots are "finitely multi-local", meaning that on a knot diagram, sums of quantities, defined by local information, determine the value of the knot invariant. The result implies the existence of Gauss diagram combinatorial formulas for finite type invariants. This article presents a simplified account of the original approach. The simplifications provide an easy generalization to the cases of pure tangles and pure braids. The associated problem on group algebras is introduced and used to prove the existence of "multi-local word formulas" for finite type invariants of pure braids.
Seifert Surface Animation	In the summer of 2006 my student Robert Barrington Leigh (1986-2006) wrote a "Live 3D" java-based web animation showing that the trefoil knot is fibered.
Braid Representatives The Multivariable Alexander Polynomial	For his summer project in 2005 my student Dan Carney wrote a Mathematica program that computes the multivariable Alexander polynomial of a link. The first and hardest step was to implement Vogel's algorithm for finding braid representatives for given links.
The Alexander-Conway Polynomial	For her summer project in 2005 my student Jana Archibald wrote a Mathematica program that computes a Gröbner basis for the higher Alexander ideals of a knot. See also her October 2007 article "The Weight System of the Multivariable Alexander Polynomial", arXiv:0710.4885.
Drawing MorseLink Presentations DT (Dowker-Thistlethwaite) Codes R-Matrix Invariants The Coloured Jones Polynomials	For his summer project in 2005 my student Siddarth Sankaran wrote a number of Mathematica programs that manipulate knots in several ways. Programs to find and draw Morse Link presentations of knots and links. Programs to convert between Gauss Codes / DT codes and other knot presentations. Programs to compute general R-Matrix invariants of knots and links. A program to compute the coloured Jones polynomial of a knot.
Bob Palais informed us of a problem with the parametrization we used in Matthew Song's java applet demonstrating the famed "Belt Trick". This applet hopefully will return here when the problem is fixed.
A Table of Virtual Knots More by Jeremy Green: JavaKh, a very fast and very general java program to compute Khovanov homology.	Here's a quote from my summer student's Jeremy Green's "About" page: Virtual knot theory is an extension of knot theory which allows for virtual crossings, which are not over or under, in a diagram. In addition to the Reidemeister moves for classical crossings, moves involving virtual crossings are allowed. For details, see Louis Kauffman, Virtual Knot Theory, Europ. J. Combinatorics (1999) 20, 663-691, arXiv:math.GT/9811028. This table of virtual knots was generated using a computer program written by Jeremy Green under the supervision of Dror Bar-Natan. The program generates a list of Gauss codes, then, via Reidemeister moves, determines which are equivalent to each other. Using an Athlon XP 2500+ with 512 MB RAM, the enumeration can be done up to 8 crossing oriented virtual knots, using 10 GB of disk, in about 24 hours. This 8 crossing enumeration is only useful for knots up to 6 crossings because of the need for higher crossing intermediates when finding relationships. This enumeration counted 725854 oriented virtual knots with up to 6 crossings, or 92800 when counting inverses and mirror images as the same knot. The program also generates all of the content on the individual pages for the virtual knots, including the drawings and the various invariants. ... To see the table, read more about it and download the programs, use the link on the left.
arXiv:0706.3680. Earlier work by Gad Naot: arXiv:math.GT/0310366, titled "On Chern-Simons theory with an inhomogeneous gauge group and BF theory knot invariants" and arXiv:math.GT/0603347, titled "On the Algebraic Structure of Bar-Natan's Universal Geometric Complex and the Geometric Structure of Khovanov Link Homology Theories".	The abstract of my student Gad Naot's PhD thesis reads as follows: We explore the complex associated to a link in the geometric formalism of Khovanov's (n=2) link homology theory, determine its exact underlying algebraic structure and find its precise universality properties for link homology functors. We present new methods of extracting all known link homology theories directly from this universal complex, and determine its relative strength as a link invariant by specifying the amount of information held within the complex. We achieve these goals by finding a complex isomorphism which reduces the complex into one in a simpler category. We introduce few tools and methods, including surface classification modulo the 4TU/S/T relations and genus generating operators, and use them to explore the relation between the geometric complex and its underlying algebraic structure. We identify the universal topological quantum field theory (TQFT) that can be used to create link homology and find that it is "smaller" than what was previously reported by Khovanov. We find new homology theories that hold a controlled amount of information relative to the known ones. The universal complex is computable efficiently using our reduction theorem. This allows us to explore the phenomenological aspects of link homology theory through the eyes of the universal complex in order to explain and unify various phenomena (such as torsion and thickness). The universal theory also enables us to state results regarding specific link homology theories derived from it. The methods developed in this thesis can be combined with other known techniques (such as link homology spectral sequences) or used in the various extensions of Khovanov link homology (such as sl3 link homology). To read his thesis, use the link on the left.
Ofer Ron remains a good friend and a former student, but our joint short note is no longer available.
Drawing Planar Diagrams	My student Emily Redelmeier wrote a Mathematica program which uses circle packings to draw knot projections and other planar diagrams.
The Planar Enumerator Page.	The proposal for Stephen Green's summer project read as follows: There is a significant number of interesting topological objects that are enumerated by decorated planar graphs modulo "simple" planar relations. These include: knots and links (enumerated by quadrivalent planar graphs with over/under crossing information indicated, modulo "Reidemeister moves"), tangles, knotted graphs, wave fronts, 2D surfaces in R4 (via their Yoshikawa-style "central frames") and many more. Some of these planar enumeration problems were studied extensively: e.g., there is vast literature on knot enumeration. Other such problems were just barely tackled: e.g., the current known enumeration of 2D surfaces in R4 fits in one page. Throughout my work in topology I have encountered a number of further enumeration problems of the same kind and wished I had had such an enumeration ready, if even rough and dirty and partial. The purpose of Stephen's project is to write a general purpose program, "The Planar Enumerator", for rough and dirty enumeration. It will have no chance of competing with the existing knot enumeration, it is likely to beat the known 2D in R^4 enumeration, and what's best, it will be completely general and whenever a planar enumeration problem will come up, it will be simple to code it in the language of "The Planar Enumerator" and get the first few pages of tables ready. The project is still at protype stage. See the link on the left.
arXiv:math.QA/0211031	Ami Haviv's PhD thesis is titled "Towards a Diagrammatic Analogue of the Reshetikhin-Turaev Link Invariants", and his abstract reads: By considering spaces of directed Jacobi diagrams, we construct a diagrammatic version of the Etingof-Kazhdan quantization of complex semisimple Lie algebras. This diagrammatic quantization is used to provide a construction of a directed version of the Kontsevich integral, denoted ZEK, in a way which is analogous to the construction of the Reshetikhin-Turaev invariants from the R-matrices of the Drinfel'd-Jimbo quantum groups. Based on this analogy, we conjecture (and prove in a restricted sense) a formula for the value of the invariant ZEK on the unknot. This formula is simpler than the Wheels formula of [BGRT:WheelsWheeling], but the precise relationship between the two is yet unknown. Here's also a brief note Ami wrote a few years ago, which implies that for the Lie algebra sl(n) there are (quite simple) cubic symmetric invariant tensors. As diagrams of the kind seen in the theory of finite type invariants cannot produce cubic symmetric invariant tensors, it follows that the diagrams to tensors map cannot be onto. This result is not new, but it's still nice to have a condensed summary: S3g.pdf, S3g.ps, S3g.tex.
Daniel Moskovich's home page arXiv:math.QA/0211223	Daniel Moskovich's reading project was to understand the various approaches to the self linking number of a space curve. He wrote an article titled "Framing and the Self-Linking Integral" on the subject (links on the left), and his abstract reads: The Gauss self-linking integral of an unframed knot is not a knot invariant, but it can be turned into an invariant by adding a correction term which requires adding extra structure to the knot. We collect the different definitions/theorems/proofs concerning this correction term, most of which are well-known (at least as folklore) and put everything together in an accessible format. We then show simply and elegantly how these approaches coincide.
Maps and Machines	Dori Eldar's MSc project was to prepare a web site (see link on the left) about "Maps and Machines". His abstract reads: In this site we study the configuration space of certain machines, all placed in the plane. Machine's configuration space is an abstract way to describe all the states the machine could take. After a short introductory to topology, we implicitly construct configuration spaces for a certain family of machines, which turn out to be, oriented surfaces of varying genus. In the third part we introduce the notion of functional linkages, which are machines who can compute polynomial functions. It can be deduced from this that to each smooth manifold M, there exists a machine with configuration space homeomorphic to a finite number of copies of M.