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Dror Bar-Natan:
Goldman-Turaev Formality from the Kontsevitch Integral
arXiv:2509.20983
pensieve
Goldman-Turaev Formality from the Kontsevitch Integral
(joint with
Zsuzsanna Dancso,
Tamara Hogan,
Jessica Liu, and
Nancy Scherich, 46 pages,
posted September 2025)
We present a new solution to the formality problem for the framed
Goldman--Turaev Lie bialgebra, constructing Goldman-Turaev homomorphic
expansions (formality isomorphisms) from the Kontsevich integral. Our
proof uses a three dimensional derivation of the Goldman-Turaev Lie
biaglebra arising from a low-degree Vassiliev quotient - the
emergent quotient - of tangles in a thickened punctured disk,
modulo a Conway skein relation. This is in contrast to Massuyeau's
2018 proof using braids. A feature of our approach is a general
conceptual framework which is applied to prove the compatibility
of the homomorphic expansion with both the Goldman bracket and the
technically challenging Turaev cobracket.
A Fast, Strong, Topologically Meaningful and Fun Knot Invariant
Theta.pdf
pensieve
A Fast, Strong, Topologically Meaningful and Fun Knot Invariant
(joint with
Roland van der Veen,
34pp,
arXiv:2509.18456).
In this paper we discuss a pair of polynomial knot invariants
$\Theta=(\Delta,\theta)$ which is:
- Theoretically and practically fast: $\Theta$ can be computed in polynomial time.
We can compute it in full on random knots with over 300 crossings,
and its evaluation at simple rational numbers on random knots with over
600 crossings.
- Strong: Its separation power is much greater than the hyperbolic volume, the
HOMFLY-PT polynomial and Khovanov homology (taken together) on knots
with up to 15 crossings (while being computable on much larger knots).
- Topologically meaningful: It likely gives a genus bound, and there are
reasons to hope that it would do more.
- Fun: Scroll to Figures 1.1--1.4, 3.1, and 6.2.
$\Delta$ is merely the Alexander polynomial. $\theta$ is almost
certainly equal to an invariant that was studied extensively by
Ohtsuki [
Oh], continuing Rozansky, Kricker, and
Garoufalidis [
Ro1,
Ro2,
Ro3,
Kr,
GR]. Yet our formulas,
proofs, and programs are much simpler and enable its computation even
on very large knots.
Appendix to \
arXiv:2504.02549
pensieve
Appendix to "Emergent Version of Drinfeld's Associator Equations" by Yusuke Kuno
(2 pages, posted April 2025,
arXiv:2504.02549)
The gist (of the appendix): It is sometimes beneficial in and
around knot theory to think of some knot strands as hard and unmoving,
or fixed, and some other strands as flexible.
Further to that, we consider the emergent quotient,
in which the flexible strands are made to be nearly transparent
to themselves and to each other - we don't quite decry that for
flexible strands (overcrossing)=(undercrossing) for that would
reduce the flexible strands to homotopy classes in the complement
of the fixed strands - yet we do mod out by relations that say that
(overcrossing)=(undercrossing) is nearly true, and so in the quotient
that remains knot theory is just barely visible, or emergent.
Computing Finite Type Invariants Efficiently
Efficiently.pdf
pensieve
arXiv:2408.15942
Computing Finite Type Invariants Efficiently
(joint with Itai Bar-Natan,
Iva Halacheva,
and
Nancy Scherich, 8 pages,
posted August 2024,
arXiv:2408.15942, to appear in Proc. Amer.
Math. Soc.)
We describe an efficient algorithm to compute finite type
invariants of type $k$ by first creating,
for a given knot $K$ with $n$ crossings, a look-up table for all
subdiagrams of $K$ of size $\lceil k/2\rceil$ indexed by dyadic intervals in
$[0,2n-1]$. Using this algorithm, any such finite type invariant can be
computed on an $n$-crossing knot in time $\sim n^{\lceil k/2\rceil},ドル a lot faster than
the previously best published bound of $n^k$.
A Perturbed-Alexander Invariant
APAI.pdf
pensieve
A Perturbed-Alexander Invariant
(joint with
Roland van der Veen,
Quantum Topology
15 (2024) 449-472,
arXiv:2206.12298).
In this note we give concise formulas, which lead to a simple and
fast computer program that computes a powerful knot invariant. This
invariant $\rho_1$ is not new, yet our formulas are by far the simplest
and fastest: given a knot we write one of the standard matrices $A$
whose determinant is its Alexander polynomial, yet instead of
computing the determinant we consider a certain quadratic expression in
the entries of $A^{-1}$. The proximity of our formulas to the Alexander
polynomial suggest that they should have a topological explanation. This
we don't have yet.
Perturbed Gaussian Generating Functions for Universal Knot Invariants
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,ドル 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.
Over then Under Tangles
OU.pdf
pensieve
Over then Under Tangles
(joint with
Zsuzsanna Dancso
and
Roland van der Veen, 35 pages,
posted July 2020, updated February 2021, Journal of Knot Theory and
its Ramifications,
32-8 (2023)
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.
Ribbon 2-Knots, 1+1=2, and Duflo's Theorem for Arbitrary Lie Algebras
arXiv:1811.08558
pensieve
Ribbon 2-Knots, 1+1=2, and Duflo's Theorem for Arbitrary Lie Algebras
(joint with
Zsuzsanna
Dancso and
Nancy
Scherich,
Algebraic & Geometric
Topology 20 (2020) 3733-3760).
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ドル+1=2$ on a
4D abacus", using the study of homomorphic expansions (aka universal
finite type invariants) for ribbon 2-knots, 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 w-knotted objects, which also
imply a relation of 2-knots with Duflo's theorem in full generality,
though via a lengthier path.
Finite Type Invariants of w-Knotted Objects IV: Some Computations
WKO4.pdf
pensieve
FreeLie.m
AwCalculus.m
Finite Type Invariants of
w-Knotted 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 "w-knotted
objects", a certain class of knotted objects in 4-dimensional
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 𝒜w, closely related to degree-completed
free Lie algebras and to degree-completed 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 𝒜w 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 knot-theoretic invariants discussed in
[WKO1]-[WKO3] and in my related
paper [KBH].
On Raoul Bott's `On Invariants of Manifold'
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.
A Note on the Unitarity Property of the Gassner Invariant
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)
3-358-17 (2015)
22-25,
arXiv:1406.7632)
We give a 3-page description of the Gassner invariant (or
representation) of braids (or pure braids), along with a description
and a proof of its unitarity property.
Finite Type Invariants of w-Knotted Objects II: Tangles, Foams and the Kashiwara-Vergne Problem
WKO2.pdf
pensieve
Finite Type Invariants of
w-Knotted Objects II: Tangles, Foams and the Kashiwara-Vergne
Problem (joint with
Zsuzsanna Dancso,
57 pages, posted May 2014 with a major update and a corrigendum on November 2023,
published in Mathematische Annalen
367 (2017) 1517-1586,
partially replaces
WKO,
arXiv:1405.1955)
This is the second in a series of papers dedicated to studying
w-knots, and more generally, w-knotted objects (w-braids, w-tangles,
etc.). These are classes of knotted objects that are wider
but weaker than their "usual" counterparts. To
get (say) w-knots from usual knots (or u-knots), one has to
allow non-planar "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 w-knotted objects a bit weaker once again.
Satoh studied several classes of w-knotted objects (under the name
"weakly-virtual") and has shown them to be closely related
to certain classes of knotted surfaces in R4.
In this article we study finite type invariants of w-tangles
and w-trivalent graphs (also referred to as w-tangled foams).
Much as the spaces A of chord diagrams for ordinary
knotted objects are related to metrized Lie algebras, the spaces
Aw of "arrow diagrams" for w-knotted objects
are related to not-necessarily-metrized Lie algebras. Many questions
concerning w-knotted objects turn out to be equivalent to questions
about Lie algebras. Most notably we find that a homomorphic
universal finite type invariant of w-foams is essentially the same
as a solution of the Kashiwara-Vergne conjecture and much of the
Alekseev-Torossian work on Drinfel'd associators and Kashiwara-Vergne
can be re-interpreted as a study of w-foams.
Finite Type Invariants of w-Knotted Objects I: w-Knots and the Alexander Polynomial
WKO1.pdf
pensieve
Finite Type Invariants of w-Knotted Objects
I: w-Knots and the Alexander Polynomial
(joint with
Zsuzsanna Dancso, 52
pages, posted May 2014, updated April 2016,
Algebraic and
Geometric Topology 16-2 (2016) 1063-1133, partially
replaces
WKO,
arXiv:1405.1956)
This is the first in a series of papers studying w-knots, and more
generally, w-knotted objects (w-braids, w-tangles, etc.). These are
classes of knotted objects which are wider but weaker
than their "usual" counterparts. To get (say) w-knots from
usual knots (or u-knots), one has to allow non-planar "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 w-knotted objects a bit weaker once again.
The group of w-braids was studied (under the name "welded
braids") by Fenn, Rimanyi and Rourke and was shown to be isomorphic
to the McCool group of "basis-conjugating" automorphisms of
a free group Fn - the smallest subgroup
of Aut(Fn) 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 R3. Satoh studied several classes
of w-knotted objects (under the name "weakly-virtual")
and has shown them to be closely related to certain classes of
knotted surfaces in R4. So w-knotted objects are
algebraically and topologically interesting.
In this article we study finite type invariants of w-braids and
w-knots. Following Berceanu and Papadima, we construct homomorphic
universal finite type invariants of w-braids. We find that the
universal finite type invariant of w-knots 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 Aw of "arrow diagrams" for w-knotted
objects are related to not-necessarily-metrized Lie algebras. Many
questions concerning w-knotted objects turn out to be equivalent to
questions about Lie algebras, and in later papers of this series we
re-interpret Alekseev-Torossian's work on Drinfel'd associators and
the Kashiwara-Vergne problem as a study of w-knotted trivalent graphs.
The true value of w-knots, 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 v-knots. We expect v-knotted objects to provide the global context
whose associated graded structure will be the
Etingof-Kazhdan theory of deformation quantization of Lie bialgebras.
2 Weight System"
align=abscenter valign=center>
arXiv:1401.0754
Proof of a Conjecture of Kulakova et al.
Related to the sl2 Weight System (joint with Huan Vo,
European Journal of Combinatorics
45 (2015) 65-70,
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
sl2
weight system on the projection of a diagram to primitive elements, is
equivalent to the Melvin-Morton-Rozansky conjecture, proved in [BNG] (MMR).
Balloons and Hoops
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 40-2 (2015) 271-329,
arXiv:1308.1721)
Balloons are two-dimensional spheres. Hoops are one dimensional
loops. Knotted Balloons and Hoops (KBH) in 4-space behave much like
the first and second fundamental groups of a topological space -
hoops can be composed as in π1, balloons
as in π2, and hoops "act" on balloons as
π1 acts on π2. We
observe that ordinary knots and tangles in 3-space map into KBH in
4-space and become amalgams of both balloons and hoops.
We give an ansatz for a tree and wheel (that is, free-Lie 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 least-wasteful in a computational sense. If you
believe in categorification, that should be a wonderful playground.
Khovanov Homology for Alternating Tangles
arXiv:1305.1695
Pensieve
Khovanov Homology for Alternating
Tangles (joint with
Hernando
Burgos-Soto, Journal of Knot Theory and its Ramifications
23-2 (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 non-split alternating link is supported in two lines. Thus our
condition is a generalization of Lee's Theorem to the case of tangles.
Meta-Monoids, Meta-Bicrossed Products, and the Alexander Polynomial
MetaMonoids.pdf
Pensieve
Meta-Monoids, Meta-Bicrossed
Products, and the Alexander Polynomial (joint with Sam
Selmani, 15 pages, posted February 2013, updated February 2014,
Journal
of Knot Theory and its Ramifications
22-10 (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.
Review of a Book by Chmutov, Duzhin, and Mostovoy
CDMReview.pdf
Pensieve
Review of a Book by Chmutov, Duzhin, and
Mostovoy (
Bull.
Amer. Math. Soc. 50 (2013) 685-690, 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 well-organized. 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 well-studied,
but hardly ever together.
Homomorphic Expansions for Knotted Trivalent Graphs
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
22-1 (2013),
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.
Pentagon and Hexagon Equations Following Furusho
arXiv:1010.0754
Pentagon and Hexagon Equations Following
Furusho (joint with
Zsuzsanna Dancso,
7 pages, posted October 2010, Proceedings
of the American Mathematical Society
140-4 (2012) 1243-1250).
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.
Some Dimensions of Spaces of Finite Type Invariants of Virtual Knots
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
20-3 (2011) 282-287,
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".
Fast Khovanov Homology Computations
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, 16-3 (2007)
243-255).
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.
Khovanov's Homology for Tangles and Cobordisms
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) 1443-1499,
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 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.
Khovanov Homology Tables
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.
Two Applications
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 7-1 (2003) 1-31, 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
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.
Categorification
paper's home
On Khovanov's Categorification of the
Jones Polynomial (posted September 2001,
Algebraic
and Geometric Topology 2-16 (2002) 337-370,
arXiv:math.QA/0201043,
updated August 2004).
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.
Bracelets
paper's home
Bracelets and the Goussarov Filtration
of the Space of Knots (posted November 26, 2001,
Invariants
of knots and 3-manifolds (Kyoto 2001), Topology and Geometry
Monographs
4, 1-12,
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.
RationalSurgery
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) 29-60,
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 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.
StatSci
paper's home
StatSci.pdf
StatSci.ps
Solving the Bible Code
Puzzle (joint with
Brendan McKay,
Gil Kalai
and Maya Bar-Hillel;
Statistical Science 14-2
(1999) 150-173)
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
AarhusIII.pdf
AarhusIII.ps
AarhusIII.tar.gz
The Århus integral of rational
homology 3-spheres III: The Relation with the Le-Murakami-Ohtsuki
Invariant (joint with
Stavros
Garoufalidis,
Lev Rozansky and
Dylan P. Thurston, Selecta Mathematica, New Series
10 (2004)
305-324,
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
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
Chance.pdf
The Torah Codes: Puzzle and
Solution (joint with Maya Bar-Hillel and
Brendan McKay
,
Chance 11-2
(1998) 13-19)
A plain-English account of some of our investigations into
"Bible codes".
Nations
paper's home
On the
Witztum-Rips-Rosenberg Sample of Nations (joint with
Brendan McKay
and Shlomo Sternberg, draft, April 1998; first edition: March
1998).
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
AarhusII.pdf
AarhusII.ps
AarhusII.tar.gz
The Århus integral of rational
homology 3-spheres II: Invariance and Universality
(joint with
Stavros
Garoufalidis,
Lev Rozansky and
Dylan P. Thurston, Selecta Mathematica, New Series
8 (2002)
341-371,
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 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.
WNP
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
AarhusI.pdf
AarhusI.ps
AarhusI.tar.gz
The Århus integral of rational
homology 3-spheres I: A highly non trivial flat connection on
S3
(joint with
Stavros
Garoufalidis,
Lev Rozansky and
Dylan P. Thurston, Selecta Mathematica, New Series
8 (2002)
315-339,
arXiv:q-alg/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 non-trivial mathematical theorems
and theories. We argue that even though non-trivial 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
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
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) 217-237,
arXiv:q-alg/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 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.
Associators
GT1.pdf
pensieve
On Associators and the
Grothendieck-Teichmuller Group I (Selecta Mathematica,
New Series
4 (1998) 183-212, June 1996, updated October
1998,
arXiv:q-alg/9606021).
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
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, New-York 1997, pp. 101-134,
arXiv:q-alg/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 (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
4CT.pdf
4CT.ps
4CT.uu
Lie Algebras and the
Four Color Theorem (Combinatorica
17-1 (1997) 43-52,
last updated October 1999,
arXiv:q-alg/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 finite-type invariants of
knots (Vassiliev invariants) and 3-manifolds.
tube
tube.pdf
tube.ps
An Elementary Proof
That All Spanning Surfaces of a Link Are
Tube-Equivalent (joint with
Jason
Fulman and
Louis H. Kauffman,
June 1995, updated March 1998, Journal of Knot Theory and its
Ramifications
7-7 (1998) 873-879).
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.
Polynomial
poly.pdf
poly.dvi
Polynomial
Invariants are Polynomial (Mathematical Research
Letters
2 (1995) 239-246,
arXiv:q-alg/9606025).
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.
Braids
glN.pdf
glN.dvi
Vassiliev and Quantum
Invariants of Braids (Proceedings of Symposia in
Applied Mathematics
51 (1996) 129-144, Amer. Math. Soc.,
arXiv:q-alg/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.
Computations
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
mmr.pdf
mmr.ps
On the
Melvin-Morton-Rozansky Conjecture (joint with
Stavros
Garoufalidis, July 1994, last updated January 1996, Inventiones
Mathematicae
125 (1996) 103-133).
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
nat.pdf
annotated:
nat@.pdf
Non-Associative
Tangles (in
Geometric topology, proceedings of the
Georgia international topology conference, W. H. Kazez, ed., 139-183,
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 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
homotopy.pdf
homotopy.tar.gz
Vassiliev Homotopy String
Link Invariants (February 1993, last updated January 1999,
Journal of Knot Theory and its Ramifications
4-1 (1995) 13-32).
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.
OnVassiliev
paper's home
OnVassiliev.pdf
On the Vassiliev Knot
Invariants (August 1992, last updated January 2007, Topology
34 (1995) 423-472).
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
thesis.pdf
Perturbative Aspects of
the Chern-Simons Topological Quantum Field Theory (Ph.D.
thesis, 109 pp, Princeton Univeristy June 1991). Contents:
PDI
the paper
Perceived Depth
Images (appeared (in a shorter form) as
Random Dot
Stereograms in The Mathematica Journal
1-3 (1991) 69-75).
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
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
NonCompact.pdf
NonCompact.dvi
Perturbative
Expansion of Chern-Simons Theory with Non-Compact Gauge
Group (joint with
Edward Witten,
Communications in Mathematical Physics
141 (1991) 423-440).
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
pcs.pdf
pcs.ps
Perturbative
Chern-Simons Theory (43 pp, April 1990, last updated
September 1995, Journal of Knot Theory and its Ramifications
4-4 (1995) 503-548).
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.
NCP
the paper
Two Examples in
Non-Commutative Probability (Foundations of Physics
19 (1989) 97-104).
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.