
My Ph.D. student Jessica Liu's
first paper's title is "A Proof of the Kashaev Signature Conjecture", and
its abstract reads:
In 2018 Kashaev introduced a diagrammatic link invariant conjectured to be twice the LevineTristram signature. If true, the conjecture would provide a simple way of computing the LevineTristram signature of a link by taking the signature of a real symmetric matrix associated with a corresponding link diagram. This paper establishes the conjecture using the Seifert surface definition of the LevineTristram signature on the disjoint union of an oriented link and its reverse. The proof also reveals yet another formula for the Alexander polynomial.Her paper is at arXiv:2311.01923. 

My masters student Daniel Martchenkov's
project's title was "Determinant Formulas for the Alexander Polynomial", and
his abstract reads:
This is a summary of various determinant formulas for the Alexander polynomial, along with their implementations in Mathematica using the KnotTheory` package.His writeup is here, and the code is here. 

My Ph.D. student Jesse Frohlich's
thesis title was "Computing the Generating Function of a Coinvariants Map", and
his abstract reads:
A wellknown source of knot invariants is representations of quasitriangular Hopf algebras (also known as quantum groups). These invariants require exponential time in the number of crossings to compute. Recent work has allowed for polynomialtime computations within the Hopf algebras themselves, using perturbed Gaussian differential operators. This thesis introduces and explores a partial expansion of the tangle invariant $Z$ introduced by BarNatan and van der Veen. We expand the use of the Hopf algebra ${\mathfrak U}({\mathfrak s\mathfrak l}^0_{2+})$ to include its space of coinvariants, providing an extension $Z^{tr}$ of $Z$ from open tangles to links.His thesis is here. 

My undergraduate student Yunchi Tang's
summer project title was "Reducing the Calculation Time of Knot Invariants", and the
first paragraph of his report reads:
In knot theory, one problem of interest is to reduce the run time of calculating knot invariants such as the Jones' polynomial. One thing we can do is to seek to reduce the amount of information that we need to process through each time we add a crossing into the calculation by permuting the order of the crossings, which often is the only information needed to calculate knot invariants.See his report here, and his programs and data files here. 

My M.Sc. student Sina Abbasi's
project title was "A Simpler Proof of McCool's Theorem", and his abstract read:
In 1947, Artin found a representation of the braid group on n strands in the automorphism group of the free group with n generators, and proved his representation is faithful. In 1986, McCool proved a theorem giving a presentation for the group of basisconjugating automorphisms of the free group with n generators. McCool's theorem can be reinterpreted as the statement that a natural analogue of Artin's representation of the braid group is a faithful representation of the pure welded braid group on n strands. However, the proof of McCool's theorem uses Whitehead's peak reduction lemma, which has no interpretation in the context of welded braids. In this paper, we apply a version of Whitehead's peak reduction lemma suited for welded braids in order to obtain a simpler proof of McCool's theorem.See his project here. 

My Ph.D. student Travis Ens'
thesis title was "On Braidors: An Analogue of the Theory of Drinfel'd Associators
for Braids in an Annulus", and his abstract read:
We develop the theory of braidors, an analogue of Drinfel'd's theory of associators in which braids in an annulus are considered rather than braids in a disk. After defining braidors and showing they exist, we prove that a braidor is defined by a single equation, an analogue of a wellknown theorem of Furusho [Furusho (2010)] in the case of associators. Next some progress towards an analogue of another key theorem, due to Drinfel'd [Drinfel'd (1991)] in the case of associators, is presented. The desired result in the annular case is that braidors can be constructed degree be degree. Integral to these results are annular versions GT_{a} and GRT_{a} of the GrothendieckTeichmuller groups GT and GRT which act faithfully and transitively on the space of braidors. We conclude by providing surprising computational evidence that there is a bijection between the space of braidors and associators and that the annular versions of the GrothendieckTeichmuller groups are in fact isomorphic to the usual versions potentially providing a new and in some ways simpler description of these important groups, although these computations rely on the unproven result to be meaningful.See his thesis here. 
 My undergraduate summerproject student David Ledvinka wrote a Mathematica program that computes the first few coefficients in the variable z of the twovariable HOMFLYPT polynomial (so in themselves they are polynomials in the variable a), following Jozef Przytycki's paper The first coefficient of Homflypt and Kauffman polynomials: Vertigan proof of polynomial complexity using dynamic programming, arXiv:1707.07733. 

My undergraduate summerproject student Calder
MortonFerguson
wrote A Visual Compansion to Hatcher's Notes on Basic 3Manifold
Topology.
See those notes as well as the accompanying Cheat Sheet. Also visit Calder's blog. 

My undergraduateproject student Andrey
Khesin
wrote two papers under my supervision!
See A Tabulation of Ribbon Knots in Tangle Form and The 250 Knots with up to 10 Crossings. Also visit Andrey at GitHub. 

My M.Sc. student Jesse Bettencourt
created the bestever explanations for why torus knots are fibered,
along with the loveliestever visualizations.
Visit Torus Knot Fibrations. (Also see a possiblyoutdated local copy and some source files). Also visit Jesse's home page! 
My Talk 
My M.Sc. student Jonathan Zung
did some lovely work on finite type invariants of doodles, but
projecttime ended before it could be written up properly. Here's the
abstract of a talk I gave
following his work at the Fields Institute in November 2014:
I will describe my former student's Jonathan Zung work on finite type invariants of "doodles", plane curves modulo the second Reidemeister move but not modulo the third. We use a definition of "finite type" different from Arnold's and more along the lines of Goussarov's "Interdependent Modifications", and come to a conjectural combinatorial description of the set of all such invariants. We then describe how to construct many such invariants (though perhaps not all) using a certain class of 2dimensional "configuration space integrals". 
Chterental's Paper 
The abstract of my Ph.D. student
Oleg Chterental's paper, titled
"Virtual Braids and Virtual Curve Diagrams", reads as follows:
This paper defines a faithful action of the virtual braid group vB_{n} on certain planar diagrams called virtual curve diagrams. Our action is similar in spirit to the Artin action of the braid group B_{n} on the free group F_{n} and it provides an easy combinatorial solution to the word problem in vB_{n}.Oleg's 2015 PhD thesis is available here. 
Our Paper 
The abstract of my joint paper with my Ph.D. student
Huan Vo, titled
"Proof of a Conjecture of Kulakova et al. Related to the
sl_{2} Weight System", reads as follows:
In this article, we show that a conjecture raised in [KLMR] (arXiv:1307.4933), which regards the coefficient of the highest term when we evaluate the sl_{2} weight system on the projection of a diagram to primitive elements, is equivalent to the MelvinMortonRozansky conjecture, proved in [BNG] (MMR). 
Our Paper. 
The abstract of my joint paper with my former M.Sc. student
Sam Selmani, titled
"MetaMonoids, MetaBicrossed Products, and the Alexander
Polynomial", reads as follows:
We introduce a new invariant of tangles along with an algebraic framework in which to understand it. We claim that the invariant contains the classical Alexander polynomial of knots and its multivariable extension to links. We argue that of the computationally efficient members of the family of Alexander invariants, it is the most meaningful. 
arXiv: 1209.4589 
The abstract of my student Karene
Chu's first paper, titled "Classification of Long Flat
Virtual Knots and a Basis for the Associated Infinitesimal
Algebra", reads as follows:
Virtual knot theory, introduced by Kauffman, is a generalization of usual knot theory which interests us because of its potential to be a topological interpretation of Etingof and Kazhdan's theory of quantization of Lie bialgebras. Usual knots inject into virtual knots, and flat virtual knots is the quotient of virtual knots which equates the real positive and negative crossings, and so in this sense is complementary to usual knot theory within virtual knot theory. We classify flat virtual pure tangles, and give bases for the "infinitesimal" algebras of two of its variants. The classification of flat virtual pure tangles can be used as an invariant on virtual pure tangles. 
Peter's Home Page. 
The abstract of my student Peter
Lee's first paper, titled "The Pure Virtual Braid Group Is
Quadratic", reads as follows:
If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra gr_{I}K need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper we give a criterion which is equivalent to gr_{I}K being quadratic. We apply this criterion to the group algebra of the pure virtual braid group (also known as the quasitriangular group), and show that the corresponding associated graded algebra is quadratic.See a video of Peter's lecture on the subject in November 2011. 
arXiv:0812.2342. 
The abstract of my student Louis
Leung's first paper reads as follows:
In 2002 Haviv gave a way of assigning Lie tensors to directed trivalent graphs. Weight systems on oriented chord idagrams modulo 6T can then be constructed from such tensors. In this paper we give explicit combinatorial formulas of weight systems using Manin triples constrcted from classical Lie algebras. We then compose these oriented weight systems with the averaging map to get weight systems on unoriented chord diagrams and show that they are the same as the ones obtained by BarNatan in 1991. In the last section we carry out calculations on certain examples. Also visit our joint 2009 paper Some Dimensions of Spaces of Finite Type Invariants of Virtual Knots. Louis' 2010 PhD thesis is Classical Lie Algebra Weight Systems of Arrow Diagrams. 
arXiv:0811.4615. 
The abstract of my student Zsuzsanna
Dancso's first paper reads as follows:
We construct an extension of the Kontsevich integral of knots to knotted trivalent graphs, which commutes with orientation switches, edge deletions, edge unzips, and connected sums. In 1997 Murakami and Ohtsuki first constructed such an extension, building on Drinfel'd's theory of associators. We construct a step by step definition, using elementary Kontsevich integral methods, to get a oneparameter family of corrections that all yield invariants well behaved under the graph operations above. Also see Zsuzsanna on YouTube! 
arXiv:0807.2600. 
The abstract of the paper Hernando
Burgos wrote as my student reads as follows:
It is a well known result from Thistlethwaite that the Jones polynomial of a nonsplit alternating link is alternating. We find the "right" generalization of this result to the case of nonsplit 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 0tangles, that is links, our condition is equivalent to simple alternation of the coefficients of the Jones polynomial. Also visit Hernando's home page! 
"Rubberband" Brunnian Links Prime Links with a NonPrime Component Identifying Knots within a List Burau's Theorem Cabling 
For her summer project in 2007 my student Iva Halacheva wrote a number of Mathematica
programs that manipulate knots in several ways.
Also visit our joint 2009 paper Some Dimensions of Spaces of Finite Type Invariants of Virtual Knots. Also visit Iva Halacheva's home page! 
arXiv:0711.4001 
Based on his master's project, my student Fionntan Roukema
wrote an article titled "GoussarovPolyakViro combinatorial formulas
for finite type invariants". His abstract reads:
Goussarov, Polyak, and Viro proved that finite type invariants of knots are "finitely multilocal", 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 "multilocal word formulas" for finite type invariants of pure braids. Also visit our joint 2009 paper Some Dimensions of Spaces of Finite Type Invariants of Virtual Knots. 
Seifert Surface Animation  In the summer of 2006 my student Robert Barrington Leigh (19862006) wrote a "Live 3D" javabased 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 AlexanderConway 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. See also her Ph.D. thesis, titled "The Multivariable Alexander Polynomial on Tangles". 
Drawing MorseLink Presentations DT (DowkerThistlethwaite) Codes RMatrix 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.

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:
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 ChernSimons theory with an inhomogeneous gauge group and BF theory knot invariants" and arXiv:math.GT/0603347, titled "On the Algebraic Structure of BarNatan'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. 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:
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
ReshetikhinTuraev Link Invariants", and his abstract reads:
By considering spaces of directed Jacobi diagrams, we construct a diagrammatic version of the EtingofKazhdan quantization of complex semisimple Lie algebras. This diagrammatic quantization is used to provide a construction of a directed version of the Kontsevich integral, denoted Z_{EK}, in a way which is analogous to the construction of the ReshetikhinTuraev invariants from the Rmatrices of the Drinfel'dJimbo quantum groups. Based on this analogy, we conjecture (and prove in a restricted sense) a formula for the value of the invariant Z_{EK} 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 SelfLinking Integral" on the subject (links on
the left), and his abstract reads:
The Gauss selflinking 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 wellknown (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. 