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I've computed Kh(T(9, 5)) and I'm happy
by
Dror Bar-Natan
University of Toronto
Coauthors: THE Computer
I'm also very tired, and there may still be bugs in the program.
Date received: February 10, 2005
3-string tangle analysis of Mu transposase
by
Isabel K. Darcy
University of Iowa
Coauthors: John Luecke, UT Austin; Mariel Vazquez, UC Berkeley
A tangle is a 3-dimensional ball containing a finite number of arcs and circles properly embedded in the 3-dimensional ball. A protein can be modeled using a 3-dimensional ball. If the protein binds to n segments of DNA, the n DNA segments can be thought of as n arcs embedded in the 3 dimensional protein ball. We will show how tangle analysis can be used to study any protein that stably binds DNA. In this talk, we will analyze an experiment by Pathania, Jayaram, Harshey (Cell, Vol. 109, 425-436) in which Mu Transposase binds 3 segments of DNA and a second protein, Cre recombinase, knots the DNA.
Date received: January 24, 2005
On Kauffman's spanning tree expansion of the Alexander polynomial
by
Oliver Dasbach
LSU
Coauthors: Charlie Frohman (University of Iowa)
Some twenty years ago, Kauffman discovered a spanning tree expansion for the Alexander polynomial. Recently, Ozsvath and Szabo used this expansion for the computation of their Floer knot homology for alternating knots. We will present some remarks on Kauffman's theory.
Date received: February 9, 2005
Polynomial and Rational knots
by
Alan Durfee
Mount Holyoke College
Coauthors: Don O'Shea, Mount Holyoke College
Abstract: A polynomial knot is defined to be a smooth embedding of R1 in R3 whose components are real polynomials. For example, the map (t5 - 10t, t4 - 4t2, t3 - 3t) represents a trefoil knot (Shastri, 1992). These knots have ends going out to infinity. Mishra, Mount Holyoke REU groups and others have found many examples of polynomial knots. Similarly a rational knot in real projective three-space is defined by rational equations. A polynomial knot completes to a rational knot, and there are many other examples.
One can show that a topological knot can be approximated by a polynomial knot of the same knot type. Given a topological knot, there are no good lower bounds for the degree of a polynomial approximation, nor are there methods for constructing examples of low degree, though past work has clarified these questions to some extent. Also the topological structure of the space of polynomial knots of a given degree is not known, though Vassiliev found this for degrees at most 4, and in degree 5 this question was partially solved by the 2002 MHC REU group.
Current research will be described and problems for future work will be indicated.
Date received: January 19, 2005
Realizations of Virtual Link Diagrams.
by
H. A. Dye
US Military Academy
A realization of a virtual link diagram is obtained by choosing over/under markings for each virtual crossing. Any realization can be obtained from some representation of the virtual link. We demonstrate that knotted realizations can be constructed from minimal genus representations that meet certain criteria.
Date received: February 6, 2005
On the Generalized Hyperbolic Volume Conjecture
by
Stavros Garoufalidis
Georgia Tech
Coauthors: Thang Le
The Generalized Hyperbolic Volume Conjecture (GHVC) states that the n-th colored Jones polynomial, evaluated at exp(2 pi i a/n), is a sequence of complex numbers that grows exponentially. Moroever, the exponential growth rate is proportional to the volume of the corresponding Dehn filling. We prove two statements: (a) the limsup in the GHVC is finite for all knots and all a. (b) for every knot K there exists a positive angle a(K) such that the GHVC holds for a in [0, a(K)). The proofs of these statements use elementary properties of state sum formulas for the colored Jones polynomial, and its recursion and its cyclotomic expansion. Viva Lou!
Date received: January 19, 2005
A Khovanov-type homology theory for graphs
by
Laure Helme-Guizon
George Washington University
Coauthors: Yongwu Rong, George Washington University, Józef Przytycki, George Washington University, Mikhail Khovanov, UC Davis.
In recent years, there has been a great deal of interests in Khovanov homology theory. For each link L, Khovanov defines a family of homology groups whose "graded" Euler characteristic is the Jones polynomial of L. These groups were constructed through a categorification process which starts with a state sum of the Jones polynomial, constructs a group for each term in the summation, and then defines boundary maps between these groups appropriately for each positive integer n.
It is natural to ask if similar categorifications can be done for other invariants with state sums.
In the first part of this talk, we establish a homology theory that categorifies the chromatic polynomial for graphs. We show our homology theory satisfies a long exact sequence which can be considered as a categorification for the well-known deletion-contraction rule for the chromatic polynomial. This exact sequence helps us to compute the homology groups of several classes of graphs. In particular, we point out that torsions do occur in the homology for some graphs. This is joint work with Yongwu Rong, George Washington University.
The second part of this talk will discuss for which graphs these homology groups have torsion. This is joint work with Yongwu Rong and Józef Przytycki, George Washington University.
The third part of this talk will discuss how to extend this construction to other algebras. This is joint work with Yongwu Rong, George Washington University and Mikhail Khovanov, UC Davis.
Date received: February 6, 2005
Linking in knot theory
by
Chun-Chung Hsieh
Math. Dept., Academia Sinica, Taiwan
Coauthors: Chun-Chung Hsieh, L.H. Kauffman
Invented by W. Massey, J. Milnor and E. Witten, linking theory is a numerical invariant of knot theory, but the explicit formulae is still missing.
In this talk, I will present the joint work with L.H. Kauffman on the first non-vanishing linkings from both Massey-Milnor and Chern-Simons-Witten aspects, aiming at the combinatorial formulae thereof.
Part I. I will define the Massey-Milnor linking and compute the combinatorial formulae explicitly.
Part II. I will talk about perturbative Chern-Simons-Witten theory and express the related linking theory explicitly in terms of Feynman diagrams.
Part III. I will show that Massey-Milnor linking is exactly the same as Chern-Simons-Witten theory.
Date received: January 8, 2005
Quadrisecants and quadrisecant approximations of knots
by
Gyo Taek Jin
Korea Advanced Institute of Science and Technology
We discuss about the minimal number of quadrisecants of knots with small crossing number. The polygonal knot inscribed in a given knot at its quadrisecant points is called the quadrisecant approximation of the knot. We conjecture that every knot and its quadrisecant approximation have the same knot type.
Date received: February 11, 2005
Robust bases and transformations of knotted cycles
by
Paul Kainen
Department of Mathematics, Georgetown University
Let G be a graph. An algebraic circuit is a set of edges with the property that every vertex is incident with an even number of edges - i.e., that induces an Eulerian subgraph. A cycle is a connected subgraph that is 2-regular. A circuit basis B is called robust provided that every cycle in the basis is a cycle and for any cycle z in G there exists an ordering (b1, b2, ¼, br) of a subset B¢ of B such that we have (i) z = b1 + b2 + ¼+ br and (ii) for every 0 < s < r , Bs+1 Ç B1 + ¼+ Bs is a nontrivial path. It follows that all of the partial sums are cycles.
If G is topologically embedded in 3-dimensional space in such a way that some cycle is knotted, then B must also contain a knotted cycle. We consider some generalizations involving various knot-theoretic invariants. We also study the effect of using a weaker notion of robustness in which the partial sums are still required to be cycles but the intersections between each new cycle and the preceding partial sum can be the union of more than two nontrivial paths. In the case of an overlap along three such paths, two unknotted cycles can give a knotted one, but this does not seem to be possible in the case that the overlap involves only two nontrivial paths.
This suggests a model for rapid topological transformation of knotted cycles involving several "active sites" and, hence, some questions regarding protein actions on polymers arise.
Date received: February 10, 2005
Generalization of a formula of Przytycki
by
Uwe Kaiser
Boise State University
J. Przytycki found an explicit formula for the image of a link in his q-homotopy skein module from the combinatorics of linking numbers. I will describe a generalization of Przytycki's formula and discuss the problems of further generalizations.
Date received: February 2, 2005
Spin Networks and Anyonic Topological Quantum Computing
by
Louis H. Kauffman
University of Illinois at Chicago
Coauthors: Samuel J. Lomonaco, UMBC
Spin networks were invented/discovered by Roger Penrose in an attempt to provide a combinatorial precursor to spacetime. In his Spin-Geometry Theorem, Penrose showed how angular properties of three dimensional space would emerge from self-interactions of large spin networks. The Penrose theory of spin networks eventually was generalized to a recoupling theory that began with the bracket polynomial skein relation rather than the Penrose binor identity. This q-deformed spin network theory has been of use in constructing SU(2)q topological quantum field theories, the Witten invariants of three manifolds, and measurement and spin-foam techniques in loop quantum gravity. Recently, Freedman, Kitaev and their collaborators have shown how braiding operators in certain topological quantum field theories are universal for quantum computation. In particular, one can focus on the topoloogical quantum field theory called Fibonacci Anyons: There are two basic particles call them 1 and 0. The only non-trivial interaction is 1 + 1 -- > 0, or 1. The corresponding recoupling theory is intricate. The braiding is non-trivial and can model quantum computation. One obtains unitary representations of the braid group that are dense in the corresponding unitary groups. The purpose of this talk is to give a simple model for the Fibonacci Anyons in terms of q = ei π/5 deformed spin networks, and to show how the structure of the model proceeds from the structure of the bracket model of the Jones polynomial. We will also discuss how these models are related to a specialization of the Durbrovnik polynomial, and how this specialization gives an invariant of virtual knots and links. The point of view of this talk allows discussion of the relationhship of quantum information theory and quantum computing with the Jones polynomial. The use of spin networks in these models suggests a deeper dialogue with quantum gravity.
Date received: February 7, 2005
The algebraic crossing number of closed braids
by
Keiko Kawamuro
Columbia University
I will show: If B_1 and B_2 are braid representatives of a knot type with the minimal braid index, then they have the same algebraic crossing number.
Date received: February 3, 2005
Categorification of the Kauffman bracket and equivariant cohomology
by
Mikhail Khovanov
UC Davis
We'll interpret Bar-Natan's link cohomology with a parameter and Rasmussen's combinatorial proof of the Milnor conjecture via equivariant cohomology.
Date received: February 4, 2005
Virtual braids and the L-move
by
Sofia Lambropoulou
National Technical Univ. Athens, Greece and Univ. de Caen, France
Coauthors: Louis Kauffman
We discuss isotopy of virtual knots and links and their topological interpretations. We then show how to derive an analogue of the Markov theorem for virtual braids using the L-moves. The L-move techniques lead to a set of local algebraic moves in the virtual braid groups that can be used for constructing invariants of virtual knots using algebraic machinery.
Date received: February 1, 2005
The slope conjecture for surgery around knots
by
Ruth Lawrence
Hebrew University, Jerusalem
Coauthors: Ofer Ron (Hebrew University)
Using Le and Habiro's techniques for computing the Ohtsuki invariant Z¥(M) of an integer homology 3-sphere M=S3K obtained by surgery around a knot K in cyclotomic form, we obtain a bound on the growth of the coefficients λn(M) of hn in Z¥(S3K), when considered as a formal power series in h=q−1.
This is consistent with the slope conjecture of Jacoby and Lawrence, namely that [(λn(M))/(λn−1(M))] is asymptotically linear in n with slope σ(M). The bound obtained on σ(S3K) is of order the square of the number of crossings in K.
Date received: February 2, 2005
On the Kauffman bracket skein modules, the Jones polynomial and the A-polynomial
by
Thang Le
Georgia Institute of technology
We calculate the Kaugmann Kracket Skein Module of complements of 2-bridge knots
and prove the AJ conjecture for a large class of 2-bridge knots.
The AJ conjecture (of Garoufalidis) says that the recursion polynomial of the colored Jones
function, when reduced q=1, is equal to the A-polynomial.
Date received: February 4, 2005
On n-punctured ball tangles
by
Xiao-Song Lin
University of California, Riverside
Coauthors: Jae-Wook Chung
We consider a class of topological objects in the 3-sphere S3 which will be called n-punctured ball tangles. Using the Kauffman bracket at A=ei pi /4, an invariant for a special type of n-punctured ball tangles is defined. The invariant F takes values in PM2×2n(Z), that is the set of 2×2n matrices over Z modulo the scalar multiplication of ±1. This invariant leads to a generalization of a theorem of D. Krebes which gives a necessary condition for a given collection of tangles to be embedded in a link in S3 disjointly. We also address the question of whether the invariant F is surjective onto PM2×2n(Z). We will show that the invariant F is surjective when n=0. When n=1, n-punctured ball tangles will also be called spherical tangles. We show that det F(S)=0 or 1 mod 4 for every spherical tangle S. Thus F is not surjective when n=1.
Date received: February 4, 2005
Blobs and Flips on Gems
by
Sóstenes Lins
Coauthors: Michele Mulazzani
In this paper we prove that two n-gems induce the same manifold if and only if they are linked by a finite sequence of gem moves. A gem move is either a blob move, consisting in the creation or cancellation of an n-dipole, or a clean flip, which is a switch of a pair of edges of the same color that thickens an h-dipole, 1 £ h £ n−1, or the inverse operation, which slims an h-dipole, 2 £ h £ n. Moreover we prove that we can reorder the gem moves, so that all the blob creations precede all clean flips which then precede all the blob cancellations. This reordering is of interest because it is an easy matter to decide whether two gems are linked by a finite sequence of clean flips. As a consequence, if a bound for the number of blob creations is established, then there exists a deterministic finite algorithm to decide whether two gems induce the same manifold or not.
Date received: February 4, 2005
The meridian maps in skein theory
by
Hugh R. Morton
University of Liverpool, UK
The meridian map is an endomorphism of the linear skein of the annulus, induced by placing an extra meridian loop around any diagram in the annulus. The eigenvalues of this map for the Homfly skein depend on two partitions, and all occur with multiplicity 1. The corresponding eigenvectors form a natural basis in many constructions of knot and manifold invariants, and they play a key role in the transition between the quantum SL(N, q) invariants of a knot and its Homfly invariants.
I shall give an account of some of the simple skein theoretic features used in constructing the eigenvectors, and their resulting properties.
Date received: January 17, 2005
A method to prove that a manifold has a complete hyperbolic structure.
by
Harriet Moser
Columbia University
Let M be a finite volume manifold where the boundary of the closure consists of a disjoint union of a finite set of tori. Then M will have a complete hyperbolic structure if there exists a simultaneous solution to the consistency and completeness equations associated with some triangulation of M. Using Newton's Method, the computer program SnapPea can approximate if this solution exists. The talk will explore a method that uses the approximation in a test that can conclusively prove that the manifold has a complete hyperbolic structure. From the consistency and completeness equations we get a function f:Cn ® Cn, and we want to know if there is a solution to f(z)=0 in the upper half plane of Cn. We use the Kantorovich Theorem, which supplies the conditions for this solution to exist in a small neighborhhod near but not including a, the approximate solution given by SnapPea. This involves finding the Lipschitz Ratio, L, for the derivative of f in this neighborhood and the supremum norm of the matrix f¢(a)−1. Implementaion and examples, including the SnapPea cusped census, will be discussed.
Date received: February 8, 2005
The twisted Alexander polynomials
by
Kunio Murasugi
University of Toronto
Coauthors: M.Hirasawa (Gakushuin University, Tokyo)
I discuss the twisted Alexander polynomials of 2-bridge knots K(a, b) associated to SL(2, F)-representations of the group of K(a, b), where F is the field of algebraic numbers over Q.
Date received: February 2, 2005
Survey on the Kauffman bracket skein modules of 3-manifolds
by
Jozef H. Przytycki
George Washington University
We describe the development of the Kauffman bracket skein modules, from the Kauffman's discovery of his bracket polynomial in 1985, to Categorification by stratified Khovanov homology of the Kauffman bracket skein module (KBSM) of FxI.
Date received: February 11, 2005
Some generalizations of quasitriangular Hopf algebras which give rise to invariants of knots and links
by
David E. Radford
University of Illinois at Chicago
Kauffman and the speaker have been interested for a very long time in generalizations of finite-dimensional Hopf algebras which account for invariants of oriented or unoriented 1-1 tangles, knots and links. Kauffman's quantum algebra accounts for the Jones polynomial. We discovered the appropriate analog, called a oriented quantum algebra, for oriented knots and links.
This talk will discuss the theory of oriented quantum algebras and closely related sructures, sketch their theory, and provide examples. There are interesting relationships between quantum algebras and oriented quantum algebras.
The Drinfel'd double of a finite-dimensional Hopf algebra has a natural oriented quantum algebra structure as do finite-dimensional representations of oriented quantum algebras. There is a very interesting oriented quantum algebra structure on the tensor product of an oriented quantum algebra with itself which is motivated by the fact that, as algebras, the Drinfel'd double D(A) of a factorizable Hopf algebra A is the tensor product of A with itself.
The talk will be a survey of joint work with Kauffman, as well as work by the speaker, on oriented quantum algebras.
Date received: February 2, 2005
Fox colorings for the number of Reidemeister moves and colored chirality
by
J. Scott Carter and Masahico Saito
U. South Alabama, U. South Florida
Coauthors: Mohamed Elhamdadi, Shin Satoh
In this talk, first, we illustrate how Fox colorings can be used to study the minimal number of Reidemeister type III moves. Functions motivated from quandle cohomology theory are used to give lower bounds for the type III moves. Second, we consider chirality of knots with Fox colorings that are also mirror images of the given colorings. We construct knots with colored chirality, and use quandle cocycle invariants as obstructions.
Date received: January 11, 2005
Disk presentations of surface-knots and -links
by
Shin Satoh
Chiba University
We introduce a new way of presenting surface-knots and -links in R4, called disk presentation. This enables us to define the disk index Δ(F) of a surface-knot or -link F. We prove that Δ(F)=2 if and only if F is a trivial S2-knot, and Δ(F)=3 if and only if F is a trivial non-orientable surface-knot with |e(F)| £ 3−χ(F), where e(F) is the normal Euler number of F and χ(F) is the Euler characteristic of F.
Date received: February 7, 2005
Rasmussen Invariant, Slice-Bennequin Inequality, and Sliceness of Knots
by
Alexander Shumakovitch
Dartmouth College
We use recently introduced Rasmussen invariant to find knots that are topologically locally-flatly slice but not smoothly slice. We note that this invariant can be used to give a combinatorial proof of the Slice-Bennequin inequality. Finally, we compute the Rasmussen invariant for quasipositive knots and show that most of our examples of non-slice knots are not quasipositive and were to the best of our knowledge unknown before.
Date received: February 10, 2005
On non-additive versions of the Kauffman-Radford reformulation of the Hennings invariant (KRH invariant.)
by
Fernando J. O. Souza
Department of Mathematics - University of Iowa
Coauthors: Part of the materials in this talk were obtained in collaboration with Louis Kauffman and David Radford.
The Kauffman-Radford reformulation of the Hennings invariant (KRH invariant) of orientable, closed 3-manifolds is defined via surgery instructions in terms of formal sums involving diagrams and Hopf-algebra objects. We will discuss variations of that definition, retaining the use of diagrammatic Hopf-algebra objects, but not the assumption that the underlying category is additive. The author and co-authors have used these versions of the KRH invariant to investigate its relationship with other invariants.
Date received: February 10, 2005
Lines and circles meeting links
by
Julia Viro
Uppsala University (Uppsala, Sweden)
We will consider low bounds for the number of lines meeting given 4 disjoint smooth closed curves in the real projective 3-space in a given cyclic order. Similarly, we estimate the number of circles meeting in a given cyclic order given 6 disjoint smooth closed curves in Euclidean 3-space. The estimations are formulated in terms of linking numbers of the curves and based on a study of a surface swept by projective lines meeting 3 given disjoint smooth closed curves and a surface swept by circles meeting 5 given disjoint smooth closed curves. These results admit generalizations to higher dimensions and more complicated patterns of intersections. Lines and circles can be replaced by configurations of curves of other kinds; linking numbers, by Vassiliev invariants.
Date received: February 4, 2005
Khovanov homology of virtual knots
by
Oleg Viro
Uppsala University (Uppsala, Sweden), Steklov Institute (St.Petersburg, Russia)
A category of alternatable virtual links is introduced and Khovanov homology of them with integer coefficients are defined. Homology with integer coefficients defined recently by Manturov for any virtual link is represented as a composition of this construction and geometric constructions.
Date received: February 4, 2005
Alexander groups of long virtual knots
by
Susan Williams
University of South Alabama
Coauthors: Daniel Silver, University of South Alabama
We define an extended Alexander group for long virtual knots, and use it ot obtain invariants of open strings (long flat virtual knots.) We show that every virtual knot is the closure of infinitely many long virtual knots. Examples of noncommuting opens strings and a ribbon concordance obstruction are given.
Date received: February 4, 2005
How the Links-Gould invariants generalise the Alexander-Conway polynomial
by
David De Wit
The University of Queensland
Coauthors: Atsushi Ishii and Jon Links
For any positive integer m, the Alexander-Conway polynomial D is obtainable as a particular one-variable `root-of-unity' reduction of the two-variable Links-Gould invariant LGm, 1. This follows as the reduction of LGm, 1 satisfies the defining skein relation of D. This is nontrivial as the reduced representation of the relevant braid generator X does not itself satisfy this relation. Instead, the key to the elegant little proof of this result involves determining the kernel of a quantum trace. We are able to evaluate this kernel from knowledge of the representation underlying LGm, 1 without having to determine X explicitly.
For positive integers m, n, strong circumstantial evidence supports a conjectured nonlinear version of this result: LGm, n reduces to a power of D. This conjecture is all the more interesting as we currently have no method of explicitly evaluating even LG2, 2 for any links other than the closures of 2-braids.
Date received: January 19, 2005
Non-linear Finite Type Invariants
by
D. N. Yetter
Kansas State University
The framed analog of the Kontsevich integral considered by Le and Murakami,
used by Le, Murakami and Ohtsuki to construct finite-type invariants of 3-manifolds,
and subsequently shown by the speaker to arise from the universal extrinsic braided
deformation of the free (symmetric) compact closed category contains more
information than is sampled by any weight system.
We suggest compelling reasons for considering certain non-linear functions in place of weight systems.
Date received: February 4, 2005
Copyright © 2005 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas.