(in approximate reverse chronological order)
PG YarnBallKnots OU UnexpectedCyclic Duflo PP1 WKO4 Bott UofG WKO2 WKO1 KLMRConj WKO KBH AltTan MetaMonoids CDMReview ktgs Furusho vDims Karoubi FastKh Cobordisms EMP KHTables TwoApplications Categorification Bracelets RationalSurgery StatSci AarhusIII Chance Nations AarhusII WNP AarhusI Wheels Associators Fundamental 4CT tube Polynomial Braids Computations MMR NAT Homotopy OnVassiliev thesis PDI Weights NonCompact pcs NCP
arXiv:2109.02057 pensieve 
Perturbed Gaussian Generating Functions for Universal Knot Invariants
(joint with Roland van der Veen,
61 pages, posted September 2021).
We introduce a new approach to universal quantum knot invariants that emphasizes generating functions instead of generators and relations. All the relevant generating functions are shown to be perturbed Gaussians of the form $Pe^G$, where $G$ is quadratic and $P$ is a suitably restricted "perturbation". After developing a calculus for such Gaussians in general we focus on the rank one invariant $Z_{\mathbb D}$ in detail. We discuss how it dominates the $\mathfrak{sl}_2$colored Jones polynomials and relates to knot genus and Whitehead doubling. In addition to being a strong knot invariant that behaves well under natural operations on tangles $Z_{\mathbb D}$ is also computable in polynomial time in the crossing number of the knot. We provide a full implementation of the invariant and provide a table in an appendix. 
YarnBallKnots.pdf pensieve 
Yarn Ball Knots and Faster Computations
(joint with Itai BarNatan, Iva Halacheva,
and Nancy Scherich, 15 pages,
posted July 2020, updated February 2021, arXiv:2108.10923)
We make use of the 3D nature of knots and links to find savings in computational complexity when computing knot invariants such as the linking number and, in general, most finite type invariants. These savings are achieved in comparison with the 2D approach to knots using knot diagrams. 
OU.pdf pensieve 
Over then Under Tangles
(joint with Zsuzsanna Dancso
and Roland van der Veen, 34 pages,
posted July 2020, updated February 2021, to appear at the Journal of Knot Theory and
its Ramifications, arXiv:2007.09828)
Brilliant wrong ideas should not be buried and forgotten. Instead, they should be mined for the gold that lies underneath the layer of wrong. In this paper we explain how "over then under tangles" lead to an easy classification of knots, and under the surface, also to some valid mathematics: a separation theorem for braids and virtual braids. We end the paper with an overview of other instances where "over then under" ideas play a role: a topological understanding of the Drinfel'd double construction of quantum group theory, the quantization of Lie bialgebras, and more. 
UnexpectedCyclic.pdf pensieve 
An Unexpected Cyclic Symmetry of $I{\mathfrak u}_n$
(joint with Roland van der Veen, 5 pages,
posted February 2020, updated February 2021, arXiv:2002.00697)
We find and discuss an unexpected (to us) order $n$ cyclic group of automorphisms of the Lie algebra $I{\mathfrak u}_n:={\mathfrak u}_n \ltimes{\mathfrak u}_n^\ast$, where ${\mathfrak u}_n$ is the Lie algebra of upper triangular $n \times n$ matrices. Our results also extend to $gl_ {n+}^\epsilon$, a "solvable approximation" of $gl_n$, as defined within. 
arXiv:1811.08558 pensieve 
Ribbon 2Knots, 1+1=2, and Duflo's Theorem for Arbitrary Lie Algebras
(joint with Zsuzsanna
Dancso and Nancy
Scherich, Algebraic & Geometric
Topology 20 (2020) 37333760).
We explain a direct topological proof for the multiplicativity of Duflo isomorphism for arbitrary finite dimensional Lie algebras, and derive the explicit formula for the Duflo map. The proof follows a series of implications, starting with "the calculation $1+1=2$ on a 4D abacus", using the study of homomorphic expansions (aka universal finite type invariants) for ribbon 2knots, and the relationship between the corresponding associated graded space of arrow diagrams and universal enveloping algebras. This complements the results of the first author, Le and Thurston, where similar arguments using a "3D abacus" and the Kontsevich Integral were used to derive Duflo's theorem for metrized Lie algebras; and results of the first two authors on finite type invariants of wknotted objects, which also imply a relation of 2knots with Duflo's theorem in full generality, though via a lengthier path. 
arXiv:1708.04853 
A Polynomial Time Knot Polynomial
(joint with Roland
van der Veen, 21 pages, posted August 2017, Proc. Amer. Math. Soc. 147 (2019) 377397).
We present the strongest known knot invariant that can be computed effectively (in polynomial time). 
WKO4.pdf pensieve FreeLie.m AwCalculus.m 
Finite Type Invariants of
wKnotted Objects IV: Some Computations
(49 pages, posted November 2015, arXiv:1511.05624)
In the previous three papers in this series, [WKO1][WKO3], Z. Dancso and I studied a certain theory of "homomorphic expansions" of "wknotted objects", a certain class of knotted objects in 4dimensional space. When all layers of interpretation are stripped off, what remains is a study of a certain number of equations written in a family of spaces 𝒜^{w}, closely related to degreecompleted free Lie algebras and to degreecompleted spaces of cyclic words. The purpose of this paper is to introduce mathematical and computational tools that enable explicit computations (up to a certain degree) in these 𝒜^{w} spaces and to use these tools to solve the said equations and verify some properties of their solutions, and as a consequence, to carry out the computation (up to a certain degree) of certain knottheoretic invariants discussed in [WKO1][WKO3] and in my related paper [KBH]. 
OnOnInvariants.pdf pensieve 
On Raoul Bott's "On Invariants of
Manifold" (2 pages, posted August 2015, in Bott's
collected works, vol. 5)
I'm not sure how to introduce a review paper. So rather than commenting on the paper as whole, I will concentrate on my subjective view of just one paragraph  a paragraph which I think I influenced and which ended up influencing me very deeply. 
UofG.pdf pensieve 
A Note on the Unitarity
Property of the Gassner Invariant (3 pages,
posted June 2014, updated August 2014, Bulletin of Chelyabinsk State
University (Mathematics, Mechanics, Informatics) 335817 (2015)
2225, arXiv:1406.7632)
We give a 3page description of the Gassner invariant (or representation) of braids (or pure braids), along with a description and a proof of its unitarity property. 
WKO2.pdf pensieve 
Finite Type Invariants of
wKnotted Objects II: Tangles, Foams and the KashiwaraVergne
Problem (joint with Zsuzsanna Dancso,
57 pages, posted May 2014, updated October 2015,
Mathematische Annalen 367 (2017) 15171586,
partially replaces WKO, arXiv:1405.1955)
This is the second in a series of papers dedicated to studying wknots, and more generally, wknotted objects (wbraids, wtangles, etc.). These are classes of knotted objects that are wider but weaker than their "usual" counterparts. To get (say) wknots from usual knots (or uknots), one has to allow nonplanar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation beyond the ordinary collection of Reidemeister moves, called the "overcrossings commute" relation, making wknotted objects a bit weaker once again. Satoh studied several classes of wknotted objects (under the name "weaklyvirtual") and has shown them to be closely related to certain classes of knotted surfaces in R^{4}. In this article we study finite type invariants of wtangles and wtrivalent graphs (also referred to as wtangled foams). Much as the spaces A of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, the spaces A^{w} of "arrow diagrams" for wknotted objects are related to notnecessarilymetrized Lie algebras. Many questions concerning wknotted objects turn out to be equivalent to questions about Lie algebras. Most notably we find that a homomorphic universal finite type invariant of wfoams is essentially the same as a solution of the KashiwaraVergne conjecture and much of the AlekseevTorossian work on Drinfel'd associators and KashiwaraVergne can be reinterpreted as a study of wfoams. 
WKO1.pdf pensieve 
Finite Type Invariants of wKnotted Objects
I: wKnots and the Alexander Polynomial
(joint with Zsuzsanna Dancso, 52
pages, posted May 2014, updated April 2016, Algebraic and
Geometric Topology 162 (2016) 10631133, partially
replaces WKO, arXiv:1405.1956)
This is the first in a series of papers studying wknots, and more generally, wknotted objects (wbraids, wtangles, etc.). These are classes of knotted objects which are wider but weaker than their "usual" counterparts. To get (say) wknots from usual knots (or uknots), one has to allow nonplanar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation beyond the ordinary collection of Reidemeister moves, called the "overcrossings commute" relation, making wknotted objects a bit weaker once again. The group of wbraids was studied (under the name "welded braids") by Fenn, Rimanyi and Rourke and was shown to be isomorphic to the McCool group of "basisconjugating" automorphisms of a free group F_{n}  the smallest subgroup of Aut(F_{n}) that contains both braids and permutations. Brendle and Hatcher, in work that traces back to Goldsmith, have shown this group to be a group of movies of flying rings in R^{3}. Satoh studied several classes of wknotted objects (under the name "weaklyvirtual") and has shown them to be closely related to certain classes of knotted surfaces in R^{4}. So wknotted objects are algebraically and topologically interesting. In this article we study finite type invariants of wbraids and wknots. Following Berceanu and Papadima, we construct homomorphic universal finite type invariants of wbraids. We find that the universal finite type invariant of wknots is more or less the Alexander polynomial (details inside). Much as the spaces A of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, we find that the spaces A^{w} of "arrow diagrams" for wknotted objects are related to notnecessarilymetrized Lie algebras. Many questions concerning wknotted objects turn out to be equivalent to questions about Lie algebras, and in later papers of this series we reinterpret AlekseevTorossian's work on Drinfel'd associators and the KashiwaraVergne problem as a study of wknotted trivalent graphs. The true value of wknots, though, is likely to emerge later, for we expect them to serve as a warmup example for what we expect will be even more interesting  the study of virtual knots, or vknots. We expect vknotted objects to provide the global context whose associated graded structure will be the EtingofKazhdan theory of deformation quantization of Lie bialgebras. 
arXiv:1401.0754 
Proof of a Conjecture of Kulakova et al.
Related to the sl_{2} Weight System (joint with Huan Vo,
European Journal of Combinatorics 45 (2015) 6570, arXiv:1401.0754).
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). 
paper's home WKO.pdf 
Finite Type Invariants of WKnotted
Objects: From Alexander to Kashiwara and Vergne.
(joint with Zsuzsanna Dancso, 100
pages, posted September 2013, updated November 2013, arXiv:1309.7155)
This paper was split in two and became the first two parts of a fourpart series (WKO1, WKO2, WKO3, WKO4). The remaining relevance of this paper's home is due to the series of videotaped lectures (wClips) that are linked there. 
paper's home KBH.pdf 
Balloons and Hoops and their Universal Finite
Type Invariant, BF Theory, and an Ultimate Alexander Invariant
(56 pages, posted August 2013, updated November 2017,
Acta
Mathematica Vietnamica 402 (2015) 271329, arXiv:1308.1721)
Balloons are twodimensional spheres. Hoops are one dimensional loops. Knotted Balloons and Hoops (KBH) in 4space behave much like the first and second fundamental groups of a topological space  hoops can be composed as in π_{1}, balloons as in π_{2}, and hoops "act" on balloons as π_{1} acts on π_{2}. We observe that ordinary knots and tangles in 3space map into KBH in 4space and become amalgams of both balloons and hoops. We give an ansatz for a tree and wheel (that is, freeLie and cyclic word) valued invariant ζ of (ribbon) KBHs in terms of the said compositions and action and we explain its relationship with finite type invariants. We speculate that ζ is a complete evaluation of the BF topological quantum field theory in 4D, though we are not sure what that means. We show that a certain "reduction and repackaging" of ζ is an "ultimate Alexander invariant" that contains the Alexander polynomial (multivariable, if you wish), has extremely good composition properties, is evaluated in a topologically meaningful way, and is leastwasteful in a computational sense. If you believe in categorification, that should be a wonderful playground. 
arXiv:1305.1695 Pensieve 
Khovanov Homology for Alternating
Tangles (joint with Hernando
BurgosSoto, Journal of Knot Theory and its Ramifications
232 (2014), 18 pages, posted May 2013,
updated March 2014, arXiv:1305.1695).
We describe a "concentration on the diagonal" condition on the Khovanov complex of tangles, show that this condition is satisfied by the Khovanov complex of the single crossing tangles, and prove that it is preserved by alternating planar algebra compositions. Hence, this condition is satisfied by the Khovanov complex of all alternating tangles. Finally, in the case of links, our condition is equivalent to a well known result which states that the Khovanov homology of a nonsplit alternating link is supported in two lines. Thus our condition is a generalization of Lee's Theorem to the case of tangles. 
MetaMonoids.pdf Pensieve 
MetaMonoids, MetaBicrossed
Products, and the Alexander Polynomial (joint with Sam
Selmani, 15 pages, posted February 2013, updated February 2014,
Journal
of Knot Theory and its Ramifications
2210 (2013), arXiv:1302.5689).
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. 
CDMReview.pdf Pensieve 
Review of a Book by Chmutov, Duzhin, and
Mostovoy (Bull.
Amer. Math. Soc. 50 (2013) 685690, posted February 2013).
Merely 30 years ago, if you had asked even the best informed mathematician about the relationship between knots and Lie algebras, she would have laughed, for there isn't and there can't be. Knots are flexible, Lie algebras are rigid. Knots are irregular, Lie algebras are symmetric. The list of knots is a lengthy mess, the collection of Lie algebras is wellorganized. Knots are useful for sailors, scouts, and hangmen, Lie algebras for navigators, engineers, and high energy physicists. Knots are blue collar, Lie algebras are white. They are as similar as worms and crystals: both wellstudied, but hardly ever together. 
paper's home ktgs.pdf 
Homomorphic Expansions for Knotted
Trivalent Graphs (joint with Zsuzsanna Dancso,
23 pages, posted March 2011, updated August 2012, Journal
of Knot Theory and Its Ramifications
221 (2013), arXiv:1103.1896).
It had been known since old times that there exists a universal finite type invariant ("an expansion") Z^{old} for Knotted Trivalent Graphs (KTGs), and that it can be chosen to intertwine between some of the standard operations on KTGs and their chorddiagrammatic counterparts (so that relative to those operations, it is "homomorphic"). Yet perhaps the most important operation on KTGs is the "edge unzip" operation, and while the behavior of Z^{old} under edge unzip is well understood, it is not plainly homomorphic as some "correction factors" appear. In this paper we present two (equivalent) ways of modifying Z^{old} into a new expansion Z, defined on "dotted Knotted Trivalent Graphs" (dKTGs), which is homomorphic with respect to a large set of operations. The first is to replace "edge unzips" by "tree connect sums", and the second involves somewhat restricting the circumstances under which edge unzips are allowed. As we shall explain, the newly defined class dKTG of knotted trivalent graphs retains all the good qualities that KTGs have  it remains firmly connected with the Drinfel'd theory of associators and it is sufficiently rich to serve as a foundation for an "Algebraic Knot Theory". As a further application, we present a simple proof of the good behavior of the LMO invariant under the Kirby II (bandslide) move. 
arXiv:1010.0754 
Pentagon and Hexagon Equations Following
Furusho (joint with Zsuzsanna Dancso,
7 pages, posted October 2010, Proceedings
of the American Mathematical Society 1404 (2012) 12431250).
In [arXiv:math/0702128] H. Furusho proves the beautiful result that of the three defining equations for associators, the pentagon implies the two hexagons (see also [Willwacher's arXiv:1009.1654]). In this note we present a simpler proof for this theorem (although our paper is less dense, and hence only slightly shorter). In particular, we package the use of algebraic geometry and GroethendieckTeichmuller groups into a useful and previously known principle, and, less significantly, we eliminate the use of spherical braids. 
paper's home 
Some Dimensions of Spaces of Finite Type
Invariants of Virtual Knots (joint with
Iva Halacheva, Louis Leung, and Fionntan Roukema,
8 pages, posted September 2009, updated January 2011, Experimental
Mathematics 203 (2011) 282287, arXiv:0909.5169).
We compute many dimensions of spaces of finite type invariants of virtual knots (of several kinds) and the dimensions of the corresponding spaces of "weight systems", finding everything to be in agreement with the conjecture that "every weight system integrates". 
paper's home 
The Karoubi Envelope and Lee's
Degeneration of Khovanov Homology (joint with Scott
Morrison, 8 pages, posted June 2006, Algebraic
& Geometric Topology 6 (2006) 14591469, arXiv:math.GT/0606542).
We give a simple proof of Lee's result from arXiv:math.GT/0210213, that the dimension of the Lee variant of the Khovanov homology of an ccomponent link is 2^{c}, regardless of the number of crossings. Our method of proof is entirely local and hence we can state a Leetype theorem for tangles as well as for knots and links. Our main tool is the "Karoubi envelope of the cobordism category", a certain enlargement of the cobordism category which is mild enough so that no information is lost yet strong enough to allow for some simplifications that are otherwise unavailable. 
paper's home 
Fast Khovanov Homology
Computations (13 pages, posted June 2006, updated May
2007, arXiv:math.GT/0606318,
Journal
of Knot Theory and Its Ramifications, 163 (2007)
243255).
We introduce a local algorithm for Khovanov Homology computations  that is, we explain how it is possible to "cancel" terms in the Khovanov complex associated with a ("local") tangle, hence canceling the many associated "global" terms in one swoosh early on. This leads to a dramatic improvement in computational efficiency. Thus our program can rapidly compute certain Khovanov homology groups that otherwise would have taken centuries to evaluate. 
paper's home Cobordism.pdf 
Khovanov's Homology for Tangles and
Cobordisms (39 pages, posted October 2004, updated April
2006, Geometry and
Topology 9 (2005) 14431499, arXiv:math.GT/0410495).
We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2knots. By staying within a world of topological pictures a little longer than in other articles on the subject, the required extension becomes essentially tautological. And then a simple application of an appropriate functor (a "TQFT") to our pictures takes them to the familiar realm of complexes of (graded) vector spaces and ordinary homological invariants. 
article's home 
Finite Type Invariants (9
pages, posted August 2004, arXiv:math.GT/0408182).
This is an overview article on finite type invariants, written for the Encyclopedia of Mathematical Physics. 
A Correction to "Groups of Ribbon Knots" by Ka Yi Ng (joint with Ofer Ron, 2 pages, posted September 2003). No longer available.  
KHTables.pdf KHTables.ps.gz 
Khovanov Homology for Knots and Links with
up to 11 Crossings (74 pages, posted May 2003, updated
August 2004).
We provide tables of the ranks of the Khovanov homology of all prime knots and links with up to 11 crossings. 
paper's home TwoApplications.pdf TwoApplications.ps arXiv:math.QA/0204311 
Two Applications of Elementary Knot
Theory to Lie Algebras and Vassiliev Invariants (joint
with Thang T. Q. Lê
and Dylan P. Thurston, Geometry
and Topology 71 (2003) 131, posted April 2002, arXiv:math.QA/0204311).
Using elementary equalities between various cables of the unknot and the Hopf link, we prove the Wheels and Wheeling conjectures of [BGRT:WheelsWheeling] and [Deligne:Letter], which give the exact Kontsevich integral of the unknot and a map intertwining two natural products on a space of diagrams. It turns out that the Wheeling map is given by the Kontsevich integral of a cut Hopf link (a bead on a wire), and its intertwining property is analogous to the computation of 1+1=2 on an abacus. The Wheels conjecture is proved from the fact that the kfold connected cover of the unknot is the unknot for all k. Along the way, we find a formula for the invariant of the general (k,l) cable of a knot. Our results can also be interpreted as a new proof of the multiplicativity of the DufloKirillov map S(g) > U(g) for metrized Lie (super)algebras g. 
paper's home 
On Khovanov's Categorification of the
Jones Polynomial (posted September 2001, Algebraic
and Geometric Topology 216 (2002) 337370, arXiv:math.QA/0201043,
updated August 2004).
The working mathematician fears complicated words but loves pictures and diagrams. We thus give a nofancyanything picturerich glimpse into Khovanov's novel construction of "the categorification of the Jones polynomial". For the same low cost we also provide some computations, including some that show that Khovanov's invariant is strictly stronger than the Jones polynomial and including a table of the values of Khovanov's invariant for all prime knots with up to 11 crossings. 
paper's home 
Bracelets and the Goussarov Filtration
of the Space of Knots (posted November 26, 2001, Invariants
of knots and 3manifolds (Kyoto 2001), Topology and Geometry
Monographs 4, 112, arXiv:math.GT/0111267).
Following Goussarov's paper "Interdependent Modifications of Links and Invariants of Finite Degree" we describe an alternative finite type theory of knots. While (as shown by Goussarov) the alternative theory turns out to be equivalent to the standard one, it nevertheless has its own share of intrinsic beauty. 
paper's home 
A Rational Surgery Formula for the LMO
Invariant (joint with Ruth Lawrence, posted
May 15, 2000, Israel Journal of
Mathematics 140 (2004) 2960, arXiv:math.GT/0007045).
We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link in S^{3}. Our main tool is a careful use of the Århus integral and the (now proven) "Wheels" and "Wheeling" conjectures of BN, Garoufalidis, Rozansky and Thurston. As steps, side benefits and asides we give explicit formulas for the values of the Kontsevich integral on the Hopf link and on Hopf chains, and for the LMO invariant of lens spaces and Seifert fibered spaces. We find that the LMO invariant does not separate lens spaces, is far from separating general Seifert fibered spaces, but does separate Seifert fibered spaces which are integral homology spheres. 
paper's home StatSci.pdf StatSci.ps 
Solving the Bible Code
Puzzle (joint with Brendan McKay, Gil Kalai
and Maya BarHillel; Statistical Science 142
(1999) 150173)
A paper of Witztum, Rips and Rosenberg in the journal Statistical Science in 1994 made the extraordinary claim that the Hebrew text of the Book of Genesis encodes events which did not occur until millennia after the text was written. In reply, we argue that Witztum, Rips and Rosenberg's case is fatally defective, indeed that their result merely reflects on the choices made in designing their experiment and collecting the data for it. We present extensive evidence in support of that conclusion. We also report on many new experiments of our own, all of which failed to detect the alleged phenomenon. 
AarhusIII.pdf AarhusIII.ps AarhusIII.tar.gz 
The Århus integral of rational
homology 3spheres III: The Relation with the LeMurakamiOhtsuki
Invariant (joint with Stavros
Garoufalidis, Lev Rozansky and
Dylan P. Thurston, Selecta Mathematica, New Series 10 (2004)
305324, arXiv:math.QA/9808013).
Continuing the work started in Part I and Part II of this series, we prove the relationship between the Århus integral and the invariant LMO defined by T.Q.T. Le, J. Murakami and T. Ohtsuki in qalg/9512002. The basic reason for the relationship is that both constructions afford an interpretation as "integrated holonomies". In the case of the Århus integral, this interpretation was the basis for everything we did in Part I and Part II. The main tool we used there was "formal Gaussian integration". For the case of the LMO invariant, we develop an interpretation of a key ingredient, the map j_{m}, as "formal negativedimensional integration". The relation between the two constructions is then an immediate corollary of the relationship between the two integration theories. 
Chance.pdf 
The Torah Codes: Puzzle and
Solution (joint with Maya BarHillel and Brendan McKay
, Chance 112
(1998) 1319)
A plainEnglish account of some of our investigations into "Bible codes". 
paper's home  On the
WitztumRipsRosenberg Sample of Nations (joint with Brendan McKay
and Shlomo Sternberg, draft, April 1998; first edition: March
1998).
We study the WitztumRipsRosenberg (WRR) sample of nations and find clear evidence that their results were obtained by selective data manipulation and are therefore invalid. Our tool is the study of variations  we vary the sample of nations in many ways, and find that the variations are almost always "worse" than the original. We argue that the only way this can be possible is if the original was "tuned" in one way or another. Finally, we show that "tuning" is a sufficiently strong process that can by itself produce results similar to WRR's. 
AarhusII.pdf AarhusII.ps AarhusII.tar.gz 
The Århus integral of rational
homology 3spheres II: Invariance and Universality
(joint with Stavros
Garoufalidis, Lev Rozansky and
Dylan P. Thurston, Selecta Mathematica, New Series 8 (2002)
341371, arXiv:math.QA/9801049).
We continue the work started in Part I, and prove the invariance and universality in the class of finite type invariants of the object defined and motivated there, namely the Århus integral of rational homology 3spheres. Our main tool in proving invariance is a translation scheme that translates statements in multivariable calculus (Gaussian integration, integration by parts, etc.) to statements about diagrams. Using this scheme the straightforward "philosophical" calculuslevel proofs of Part I become straightforward honest diagramlevel proofs here. The universality proof is standard and utilizes a simple "locality" property of the Kontsevich integral. 
paper's home  Equidistant Letter
Sequences in Tolstoy's "War and Peace" (joint with Brendan McKay,
draft, December 1997; first edition: September 1997).
In [WRR1], Witztum, Rips and Rosenberg found a surprising correlation between famous rabbis and their dates of birth and death, as they appear as equidistant letter sequences in the Book of Genesis. We make a smaller or equal number of mistakes, and find the same phenomenon in Tolstoy's eternal creation "War and Peace". 
AarhusI.pdf AarhusI.ps AarhusI.tar.gz 
The Århus integral of rational
homology 3spheres I: A highly non trivial flat connection on
S^{3}
(joint with Stavros
Garoufalidis, Lev Rozansky and
Dylan P. Thurston, Selecta Mathematica, New Series 8 (2002)
315339, arXiv:qalg/9706004).
Path integrals don't really exist, but it is very useful to dream that they do and figure out the consequences. Apart from describing much of the physical world as we now know it, these dreams also lead to some highly nontrivial mathematical theorems and theories. We argue that even though nontrivial flat connections on S^{3} don't really exist, it is beneficial to dream that one exists (and, in fact, that it comes from the nonexistent ChernSimons path integral). Dreaming the right way, we are led to a rigorous construction of a universal finitetype invariant of rational homology spheres. We show that this invariant recovers the Rozansky and Ohtsuki invariants and that it is essentially equal to the LMO (LeMurakamiOhtsuki) invariant. This is part I of a 4part series, containing the introductions and answers to some frequently asked questions. Theorems are stated but not proved in this part, and it can be viewed as a "research announcement". Part II of this series is titled "Invariance and Universality", Part III is titled "The Relation with the LeMurakamiOhtsuki Invariant", and part IV will be titled "The Relation with the Rozansky and Ohtsuki Invariants". 
Wheels.pdf Wheels.uu 
Wheels, Wheeling, and the Kontsevich
Integral of the Unknot (joint with Stavros
Garoufalidis, Lev Rozansky and
Dylan P. Thurston, posted March 1997, Israel Journal of Mathematics
119 (2000) 217237, arXiv:qalg/9703025).
We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne) for the relation between the two natural products on the space of unitrivalent diagrams. The two formulas use the related notions of "Wheels" and "Wheeling". We prove these formulas "on the level of Lie algebras" using standard techniques from the theory of Vassiliev invariants and the theory of Lie algebras. 
GT1.pdf pensieve 
On Associators and the
GrothendieckTeichmuller Group I (Selecta Mathematica,
New Series 4 (1998) 183212, June 1996, updated October
1998, arXiv:qalg/9606021).
We present a formalism within which the relationship (discovered by Drinfel'd) between associators (for quasitriangular quasiHopf algebras) and (a variant of) the GrothendieckTeichmuller group becomes simple and natural, leading to a great simplification of Drinfel'd's original work. In particular, we reprove that rational associators exist and can be constructed iteratively. 
Fundamental.pdf Fundamental.ps Fundamental.uu  The Fundamental
Theorem of Vassiliev Invariants (joint with Alexander
Stoimenow, Geometry and Physics, (J.E. Andersen, J. Dupont, H.
Pedersen, and A. Swann, eds.), lecture notes in pure and applied
mathematics 184, Marcel Dekker, NewYork 1997, pp. 101134, arXiv:qalg/9702009).
An exposition of four approaches to the proof of "The Fundamental Theorem of Vassiliev Invariants", saying that every weight system can be integrated to an invariant. We argue that each of these approaches (topologicalcombinatorial, geometric, physical, and algebraic) is, in some sense, wrong. The first and most natural approach simply fails, but while the other three succeed, they still appear unnatural. We express our hopes that these difficulties are an indication that there's something hiding around there, waiting to be discovered. The only new mathematics in this paper is a repackaging of Hutchings' topologicalcombinatorial argument in terms of the snake lemma. 
4CT.pdf 4CT.ps 4CT.uu  Lie Algebras and the
Four Color Theorem (Combinatorica 171 (1997) 4352,
last updated October 1999, arXiv:qalg/9606016).
Contains an appealing statement about Lie algebras that is equivalent to the Four Color Theorem. The notions appearing in the statement also appear in the theory of finitetype invariants of knots (Vassiliev invariants) and 3manifolds. 
tube.pdf tube.ps  An Elementary Proof
That All Spanning Surfaces of a Link Are
TubeEquivalent (joint with Jason
Fulman and Louis H. Kauffman,
June 1995, updated March 1998, Journal of Knot Theory and its
Ramifications 77 (1998) 873879).
The standard proof that the potential function provides a model for the AlexanderConway polynomial depends on the fact that all Seifert surfaces of a link are tubeequivalent. Proofs that all Seifert surfaces of a link are tubeequivalent use machinery such as the ThomPontrjagin construction. Here we present an elementary geometric argument in the case of links in the three dimensional sphere which allows one to visualize the additions and removals of tubes. 
poly.pdf poly.dvi  Polynomial
Invariants are Polynomial (Mathematical Research
Letters 2 (1995) 239246, arXiv:qalg/9606025).
Contains a proof that (as conjectured by Lin and Wang [arXiv:dgga/9411015] when a Vassiliev invariant of type m is evaluated on a knot projection having n crossings, the result is bounded by a constant times n^{m}. Thus the well known analogy between Vassiliev invariants and polynomials justifies (well, at least explains) the odd title of this note. 
glN.pdf glN.dvi  Vassiliev and Quantum
Invariants of Braids (Proceedings of Symposia in
Applied Mathematics 51 (1996) 129144, Amer. Math. Soc., arXiv:qalg/9607001).
Contains a proof of the fact that all Vassiliev invariants of braids come (in the natural sense) from gl(N) and its representations, and thus, in the light the fact that Vassiliev invariants separate braids (see my paper Vassiliev Homotopy String Link Invariants ), the gl(N) invariants separate braids. A nice corollary of that is that every Vassiliev invariant of braids extends to a Vassiliev invariant (of the same degree) of string links. 
table.pdf table.dvi  Some Computations Related
to Vassiliev Invariants (18 pp, last updated May 5, 1996,
available online, not meant for publication).
Contains tables of dimensions of over 500 spaces of Chinese Characters, as well as some tables of dimensions of spaces of Vassiliev invariants of knots, braids, and string links. Also contains the decompositions into irreducibles of the representations of the symmetric groups naturally associated with Chinese Characters. The data summarized in these tables is available in a mathematica readable format (also viewable as plain text) in a single data file, table.m. The undocumented source code is at Pensieve: Projects: JacobiDiagrams. 
mmr.pdf mmr.ps  On the
MelvinMortonRozansky Conjecture (joint with Stavros
Garoufalidis, July 1994, last updated January 1996, Inventiones
Mathematicae 125 (1996) 103133).
We prove a conjecture made by Melvin and Morton, saying that a certain specialization of the colored Jones polynomial is equal to the inverse of the Conway polynomial (in particular, the Conway polynomial is computable from the Jones polynomial and its cablings, something that was not known before). Later, Rozansky gave (here) a nonrigorous ChernSimons path integral "proof" of that conjecture, which suggests a generalization (which we also prove) to arbitrary Lie algebras. Rozansky's techniques do not appear to be related to ours. Other reasons to read our paper:

nat.pdf annotated: nat@.pdf  NonAssociative
Tangles (in Geometric topology, proceedings of the
Georgia international topology conference, W. H. Kazez, ed., 139183,
Amer. Math. Soc. and International Press, Providence, 1997).
We give a first completely combinatorial construction of a universal Vassiliev invariant, along lines suggested by Drinfel'd's work on quasiHopf algebras (previous papers on the subject did not give a combinatorial construction of an associator Phi). We describe a mathematica program implementing our algorithm, compute an associator up to degree 7, and compute our invariant in a few simple cases. 
homotopy.pdf homotopy.tar.gz  Vassiliev Homotopy String
Link Invariants (February 1993, last updated January 1999,
Journal of Knot Theory and its Ramifications 41 (1995) 1332).
I show that the main conjectures of On the Vassiliev Knot Invariants become theorems when the attention is restricted to string links considered only up to homotopy. That is, the corresponding map into surfaces is injective (so all homotopy invariants come from surfaces), and Vassiliev homotopy invariants separate homotopy string links. The later result is proven by showing that the Milnor mu invariants are Vassiliev invariants. Along the way we also find that Vassiliev invariants of braids separate braids. 
paper's home OnVassiliev.pdf  On the Vassiliev Knot
Invariants (August 1992, last updated January 2007, Topology
34 (1995) 423472).
An introduction to Vassiliev invariants. Contains the definition, proofs that the various knot polynomials are Vassiliev invariants (appropriately parametrized and expanded), the basic constructions (of weight systems from Vassiliev invariants and from Lie algebras), a discussion of the Hopf algebra of chord diagrams, The Kontsevich integral proving that every weight system comes from an invariant, the diagrammatic PBW theorem and Chinese characters, the map into marked surfaces, an analysis of the space of weight systems coming from that map (exactly all classical algebras), and some more. If you're a newcomer to the field and you're asking me, that's the paper to read! 
thesis.pdf  Perturbative Aspects of
the ChernSimons Topological Quantum Field Theory (Ph.D.
thesis, 109 pp, Princeton Univeristy June 1991). Contents:

the paper  Perceived Depth
Images (appeared (in a shorter form) as Random Dot
Stereograms in The Mathematica Journal 13 (1991) 6975).
Describes a Mathematica package for creating perceived depth images  these things that look 3D when you look at them with your eyes crossed. For the mathematica package itself, click here. For a primitive but working 9 line version of that package, click here. 
weights.pdf weights.ps  Weights of Feynman
Diagrams and the Vassiliev Knot Invariants (22 pp,
February 1991, last updated June 1995).
My first paper on Vassiliev invariants, the first place where the relation between Vassiliev invariants and Lie algebras was noticed, and the first place where it was shown that there are more than finitely many Vassiliev invariants (by showing that the coefficients of the Conway polynomial are Vassiliev invariants). This paper is almost entirely a subset of my On the Vassiliev Knot Invariants. Perhaps the only thing which is still of interest in it is an algorithm for computing the weight systems associated with the symplectic groups. 
NonCompact.pdf NonCompact.dvi  Perturbative
Expansion of ChernSimons Theory with NonCompact Gauge
Group (joint with Edward Witten,
Communications in Mathematical Physics 141 (1991) 423440).
A discussion of the semiclassical approximation for ChernSimons theory with a noncompact gauge group. After finding the correct gauge fixing, we discuss the somewhat nonstandard eta invariant that enters the computation of the phase of the path integral, and a certain anomaly related to it. 
pcs.pdf pcs.ps  Perturbative
ChernSimons Theory (43 pp, April 1990, last updated
September 1995, Journal of Knot Theory and its Ramifications
44 (1995) 503548).
Contains an introduction to perturbation theory in the context of ChernSimons theory and knots, a discussion of the first order perturbation theory (linking, selflinking, and the torsionrelated anomaly that forces the introduction of framings), a proof that the second order pertubation theory converges and yields a familiar knot invariant whose reduction mod 2 is the arf invariant, and a discussion of what is expected to happen at higher orders. 
the paper  Two Examples in
NonCommutative Probability (Foundations of Physics
19 (1989) 97104).
Mainly an exposition of the Bell inequality from the point of view of noncommutative (quantum) probability. Also contains a short discussion of the Heisenberg uncertainty principle from the same point of view. 