paper's home
In Preparation.
arXiv:1103.1896
It had been known since old times that there exists a universal finite type invariant ("an expansion") Zold for Knotted Trivalent Graphs (KTGs), and that it can be chosen to intertwine between some of the standard operations on KTGs and their chord-diagrammatic 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 Zold 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 Zold 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 (band-slide) move.
arXiv:1010.0754
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 Groethendieck-Teichmuller groups into a useful and previously known principle, and, less significantly, we eliminate the use of spherical braids.
paper's home
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
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 c-component link is 2c, regardless of the number of crossings. Our method of proof is entirely local and hence we can state a Lee-type 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
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
We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. 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
This is an overview article on finite type invariants, written for the Encyclopedia of Mathematical Physics.
KHTables.pdf
KHTables.ps.gz
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
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 k-fold 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 Duflo-Kirillov map S(g) --> U(g) for metrized Lie (super-)algebras g.
paper's home
The working mathematician fears complicated words but loves pictures and diagrams. We thus give a no-fancy-anything picture-rich 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
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
We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link in S3. Our main tool is a careful use of the Århus integral and the (now proven) "Wheels" and "Wheeling" conjectures of B-N, 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
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
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 q-alg/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 jm, as "formal negative-dimensional integration". The relation between the two constructions is then an immediate corollary of the relationship between the two integration theories.
Chance.pdf
A plain-English account of some of our investigations into "Bible codes".
paper's home
We study the Witztum-Rips-Rosenberg (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
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 3-spheres. Our main tool in proving invariance is a translation scheme that translates statements in multi-variable calculus (Gaussian integration, integration by parts, etc.) to statements about diagrams. Using this scheme the straight-forward "philosophical" calculus-level proofs of Part I become straight-forward honest diagram-level proofs here. The universality proof is standard and utilizes a simple "locality" property of the Kontsevich integral.
paper's home
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
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 non-trivial mathematical theorems and theories. We argue that even though non-trivial flat connections on S3 don't really exist, it is beneficial to dream that one exists (and, in fact, that it comes from the non-existent Chern-Simons path integral). Dreaming the right way, we are led to a rigorous construction of a universal finite-type 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 (Le-Murakami-Ohtsuki) invariant.
This is part I of a 4-part 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 Le-Murakami-Ohtsuki Invariant", and part IV will be titled "The Relation with the Rozansky and Ohtsuki Invariants".
Wheels.pdf
Wheels.uu
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 uni-trivalent 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
GT1.uu
We present a formalism within which the relationship (discovered by Drinfel'd) between associators (for quasi-triangular quasi-Hopf algebras) and (a variant of) the Grothendieck-Teichmuller group becomes simple and natural, leading to a great simplification of Drinfel'd's original work. In particular, we re-prove that rational associators exist and can be constructed iteratively.
Fundamental.pdf
Fundamental.ps
Fundamental.uu
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 (topological-combinatorial, 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' topological-combinatorial argument in terms of the snake lemma.
4CT.pdf
4CT.ps 4CT.uu
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 finite-type invariants of knots (Vassiliev invariants) and 3-manifolds.
tube.pdf
tube.ps
The standard proof that the potential function provides a model for the Alexander-Conway polynomial depends on the fact that all Seifert surfaces of a link are tube-equivalent. Proofs that all Seifert surfaces of a link are tube-equivalent use machinery such as the Thom-Pontrjagin 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
Contains a proof that (as conjectured by Lin and Wang [arXiv:dg-ga/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 nm. 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
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
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 rather large data file (>800Kb compressed, >9Mb in full!), table.m. (If you're using Netscape, you may want to click SHIFT-LEFT on table.m when downloading it, to prevent automatic decompression).
mmr.pdf
mmr.ps
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 non-rigorous Chern-Simons 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:
- We prove the theorem on the level of weight systems and then use a general principle to show that in some cases (ours included), this is sufficient. We expect that there will be other applications to this method of proof.
- We use a nice (and new) relation between the weight system of the Conway polynomial and the intersection graph of a chord diagram.
- We discuss an amusing (and new) relation between immanants and the algebra generated by the coefficients of the Conway polynomial.
- We give a general formula for the weight system corresponding to the Lie algebra su(2) in an arbitrary representation.
nat.pdf
nat.ps
nat.dvi
We give a first completely combinatorial construction of a universal Vassiliev invariant, along lines suggested by Drinfel'd's work on quasi-Hopf 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.ps
homotopy.dvi
homotopy.uu
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
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
- Introduction, The Feynman rules for Chern-Simons theory in flat space.
- My paper Perturbative Chern-Simons Theory, in a slightly older format.
- A general method for translating the BRST argument to the level of Feynman diagrams.
- The first "formal" proof (and the only one fully written up) that Chern-Simons perturbation theory gives knot invariants to all orders. The proof is on the formal algebraic level and does not deal (unfortunately) with divergencies. Modulo this point, it is also the first proof that every weight system can be integrated to a Vassiliev invariant.
- My paper Weights of Feynman Diagrams and the Vassiliev Knot Invariants.
- A somewhat different derivation of the results in my paper with Witten Perturbative Expansion of Chern-Simons Theory with Non-Compact Gauge Group.
the paper
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
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
A discussion of the semi-classical approximation for Chern-Simons theory with a non-compact gauge group. After finding the correct gauge fixing, we discuss the somewhat non-standard eta invariant that enters the computation of the phase of the path integral, and a certain anomaly related to it.
pcs.pdf
pcs.ps
Contains an introduction to perturbation theory in the context of Chern-Simons theory and knots, a discussion of the first order perturbation theory (linking, self-linking, and the torsion-related 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
Mainly an exposition of the Bell inequality from the point of view of non-commutative (quantum) probability. Also contains a short discussion of the Heisenberg uncertainty principle from the same point of view.
