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The alternation number and the Rasmussen invariant
by
Tetsuya Abe
Osaka City University
The alternation number of a knot is an obstruction to the knot being alternating, which is defined to be the minimal number of crossing changes needed to deform the knot into an alternating knot. We give a lower bound for the alternation number of a knot by using the Rasmussen's s-invariant and the signature of a knot.
Date received: December 1, 2008
Complementary Regions of Knot and Link Projections
by
Colin Adams
Williams College
Coauthors: Reiko Shinjo, Kokoro Tanaka
A strictly increasing sequence of integers is said to be universal if every knot can be realized by a projection, the complementary regions of which are n-gons, for values of n that occur in the list. We prove that (2, n, n+1, ...) for n ≥ 3 and (3, n, n+1, ...) for n ≥ 4 are universal. Moreover, (3, 4, n) is universal for all n ≥ 5, and (2, 4, 5) is universal. We also consider sequences for n-component links.
Date received: November 17, 2008
Braid group rerresentations arising from the affine Grassmannian and factorization spaces
by
Kenji Aragane
Osaka City University
There are well known constructions of representations of braid groups by flat fibre bundles on configuration spaces. In this construction, flat structures are essential. We begin by constructing fibrations with flat structures on the projective space, which are related to the categorification of knot invariants, and we explain the relationship between our fibrations and factorization spaces, introduced by Beilinson and Drinfeld as a nonlinear version of vertex algebras.
Our construction of representations are geometric and closely related to the geometric Langlands correspondence.
Date received: December 1, 2008
The Multivariable Alexander Polynomial on Tangles.
by
Jana Archibald
University of Toronto
I will give an extension of the multivariable Alexander polynomial to tangles. The good behaviour of this invariant under tangle composition will give us ease of computation and a uniform proof of many Alexander relations.
Date received: December 2, 2008
A Construction on Tangles Extending the A-Polynomial
by
John Armstrong
Tulane University
The A-polynomial (Cooper-Culler-Gillet-Long-Shalen) is an interesting invariant of knots defined by considering representations of the knot complement and how they restrict to the complement's boundary. We give a construction which generalizes this procedure and extends it to tangles. It is our hope that this will lead to an algorithm for calculating the A-polynomial of any knot.
Date received: January 12, 2009
Singular link Floer Homology
by
Benjamin Audoux
Unige (University of Geneva)
There are different ways to study the Alexander polynomial Δ. It can be done using Vassiliev theory, i.e. the generalization of Δ to singular links, but also using its categorification, i.e. Heegaard-Floer homology which is a sequence of homology groups with Δ as graded Euler caracteristic. The question of a possible relation between these two constructions is then naturally raised. In my talk, I will give a generalization of Heegaard-Floer homology to singular links which categorifies the Vasiliev iterative formula and which satisfies some "good" acyclicity conditions.
Date received: November 19, 2008
Pull Back Relation for Non-Spherical Knots
by
Vincent Blanloeil
I.R.M.A. Louis Pasteur University of Strasbourg
Coauthors: Y. Matsumoto, O. Saeki
We introduce a new relation for high dimensional non-spherical knots, which is motivated by the codimension two surgery theory: a knot is a pull back of another knot if the former is obtained as the inverse image of the latter by a certain degree one map between the ambient spheres. This relation defines a partial order for (2n-1)-dimensional simple fibered knots for n ≥ 3. We give some related results concerning cobordisms and isotopies of knots.
Date received: December 1, 2008
An explicit sphere eversion - 50 years after Smale's Theorem
by
J Scott Carter
University of South Alabama
The intricate details of an eversion that is related to the Froisart-Morin eversion are exposed via a variety of techniques. The Carter-Reiger-Saito movie-moves (excluding those that involve branch points) are used in a step-by-step process to give movies, charts, and the decket-sets of the eversion. The double-point set and the fold set of the viewing projection are calculated, and interesting eversion questions are asked.
Date received: December 2, 2008
Quasi-tree expansion for the Bollobas-Riordan-Tutte polynomial
by
Abhijit Champanerkar
College of Staten Island, CUNY
Coauthors: Ilya Kofman and Neal Stoltzfus
Bollobas and Riordan introduced a three-variable polynomial extending the Tutte polynomial to oriented ribbon graphs, which are multi-graphs embedded in oriented surfaces, such that complementary regions (faces) are discs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. By generalizing Tutte's concept of activity to quasi-trees, we extend the spanning tree expansion of the Tutte polynomial to a quasi-tree expansion of the Bollobas-Riordan-Tutte polynomial.
Paper reference: arXiv:0705.3458
Date received: November 25, 2008
Obstructions to slicing Bing doubles
by
Cornelia Van Cott
University of San Francisco
Much recent attention has focused on characterizing when a Bing double is slice. We will discuss recent progress toward achieving this goal using a new tool called covering link calculus.
Date received: November 20, 2008
The Arrow Polynomial
by
Heather A. Dye
McKendree University
Coauthors: Louis H. Kauffman
We introduce the arrow polynomial, an invariant of virtual link diagrams. This polynomial is an oriented version of the bracket polynomial; associated with each surviving state of the arrow polynomial is a k-degree. The maximum k-degree of the arrow polynomial determines a lower bound on the virtual crossing number of a virtual link.
Paper reference: arXiv:0810.3858
Date received: October 8, 2008
Small braids with large ultra summit sets
by
Ivan Dynnikov
Moscow State University
Coauthors: Maksim Prasolov
I will present a sequence of braids, the nth one having n strands and n-1 crossings, whose ultra summit set is exponentially large in n. All braids in the sequence are pseudo-Anosov and rigid.
This gives a negative answer to a question raised recently by J.Birman, V.Gebhardt, and J.Gonz´alez-Meneses.
Date received: December 26, 2008
Finite type invariants of links and surfaces in 3-space
by
Michael Eisermann
Institut Fourier, Université Grenoble I
Slice and ribbon knots are a classical subject of knot theory ever since the seminal work of Fox and Milnor 50 years ago. Contrary to the Alexander polynomial, the Jones polynomial does not seem to reflect these topological properties. In this talk I present some results towards understanding the Jones polynomial of ribbon links, and more generally of immersed ribbon surfaces in 3-space. The right point of view is the power series expansion at t=-1 instead of t=1 as usual. The coefficients, beginning with the determinant in degree 0, are not of finite type in the sense of Vassiliev-Goussarov, but they turn out to be of finite type in the appropriate sense for (embedded or immersed) surfaces bounding links in 3-space. Motivated by this example, I shall sketch the theory of surface invariants of finite type. The aim is to reconcile quantum invariants with the classical setting of links and surfaces, and to naturally place some classical invariants of algebraic topology into the framework of an extended finite type theory.
Paper reference: arXiv:0802.2287
Date received: November 9, 2008
How to categorify dynamical zeta functions
by
Alexander Fel'shtyn
University of Szczecin and Boise State University
A programm of a categorification a la Khovanov of Weil type dynamical zeta functions is proposed.
(Categorification of the Weil zeta function)
Let φ: Σ→ Σ be a symplectomorphism of a compact surface.
Then
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There are intriguing questions about categorifications of arithmetic L functions.
Date received: October 22, 2008
Plumbers' knots and unstable Vassiliev theory
by
Chad Giusti
University of Oregon
We begin by constructing the spaces of plumbers' knots, which are piecewise linear with all sticks parallel to the axes. These knots are closely related to lattice knots and provide a new version of finite complexity knot theory which gives rise to classical knot theory. The combinatorial structure of these spaces allows us to construct an algorithm to enumerate knot types with a bounded number of moves. An implementation of this algorithm has demonstrated that, for example, there are seven components of the space of plumbers' knots with five moves, though they all fall into one of three topological types.
We next extend the notion of Vassiliev derivatives to singular knots with singularities other than collections of isolated double-points. We then import Vassiliev's original techniques to the plumber's knots setting, constructing an unstable Vassiliev spectral sequence which is compatible with stabilization (subdivision of segments). This result opens the door to constructing new Vassiliev-style knot invariants and/or seeing the strength of finite-type invariants once we understand the behavior of Vassiliev derivatives under stabilization.
Date received: October 21, 2008
Non-quasi-alternating Montesinos links, and lens space surgeries on knots
by
Josh Greene
Princeton University
I will discuss some examples of links whose non-quasi-alternating-ness can be established by the study of negative definite 4-manifolds, by contrast to homological width. In particular, some of these examples have thin knot Floer and reduced Khovanov homology. Whether their odd Khovanov homology is thin remains to be seen (though I may know more when I speak). As a related application, I will show that if p is positive integer, and p-surgery on a knot gives a lens space, then the knot genus is bounded above by (p -√(cp))/2 for some absolute constant c between 1.5 and 4. For p large, this improves on a conjectured bound by Goda and Teragaito.
Date received: December 29, 2008
Pseudo diagrams of knots, links and spatial graphs
by
Ryo Hanaki
Graduate School of Education, Waseda University
A diagram of a knot, a link or a spatial graph is a projection of it into the 2-sphere with over/under information at the every double points. In this talk we introduce a notion of a pseudo diagram which is a projection with over/under information at some double points of it. We discuss whether or not we can determine the triviality or the non-triviality of the knot from the pseudo diagram.
Date received: November 29, 2008
Semiquandles and flat virtuals
by
Allison Henrich
Oberlin College
Coauthors: Sam Nelson
Flat virtual knots have recently emerged as an interesting class of knots. Also referred to as homotopy classes of virtual knots or virtual strings, these knots can be used to help us classify ordinary virtual knots. We introduce an algebraic structure called a semiquandle that yields several new invariants for flat virtual knots. We also introduce the notion of singular semiquandle. The singular semiquandle allows us to generalize to invariants for flat singular virtual knots. As the terminology suggests, these structures are related to the biquandles that give us invariants for classical knots. This is joint work with Sam Nelson.
Date received: October 23, 2008
A categorification of Hecke algebras
by
Alex Hoffnung
University of California, Riverside
Coauthors: John Baez
Given a Dynkin diagram and the finite field Fq, where q is a prime power, we get a finite algebraic group Gq. We will show how to construct a categorification of the Hecke algebra H(Gq) associated to this data. This is an example of the Baez/Dolan program of "Groupoidification", a method of promoting vector spaces to groupoids and linear operators to spans of groupoids. For example, given the A2 Dynkin diagram, for which Gq = SL(3, q), the spans over the Gq-set of complete flags in Fq3 encode the relations of the Hecke algebra associated to SL(3, q). Further, we will see how the categorified Yang-Baxter equation is derived from incidence relations in projective plane geometry.
Date received: November 24, 2008
Almost alternating knots producing an alternating knot
by
Sumiko Horiuchi
Tokyo Woman's Christian University , Japan
Coauthors: Yoshiyuki Ohyama (Tokyo Woman's Christian University)
Adams et al. introduce the notion of almost alternating links; nonalternating links which have a projection whose one crossing change yields an alternating projection. For an alternating knot K, we consider the number Alm(K) of almost alternating knots which have a projection whose one crossing change yields K. We show that for any given natural number n, there is an alternating knot K with Alm(K) > n.
Date received: November 21, 2008
The planar algebra associated to an alternating link and its Jones Polynomial.
by
Stella Huerfano
University of Bogota, Columbia
The Jones polynomial of a non-split alternating link is alternating. A generalization of this result to the case of non-split alternating
tangles is presented. The Jones polynomial of tangles is valued in a certain skein module, where the alternating condition for tangles can be described.
It has been shown by Hernando Burgos (see his PhD thesis) that this
condition is "preserved" by the Jones polynomial associated to
the diagrams of the single positive and negative crossings,
and it is also preserved by appropriately chosen "alternating"
planar algebra compositions.
As a result, we demonstrate that this condition, for links, reduces to the alternation of the coefficients of the Jones polynomial.
Date received: January 12, 2009
Surfaces associated to a knotted space curve
by
Barbara Jablonska
TU Berlin
From any smooth knot K in 3-space, we derive three surfaces with maps to S^2. These represent certain geometric features of the knot diagrams obtained as the orthogonal projections of K in all possible directions. It turns out that these surfaces have the same boundary curves and fold lines on S^2. One can cut them apart and reglue in various different manners obtaining in each case a different compact 2-manifold with a map to S^2. The degrees of these maps yield some interesting results. E.g. in one case the degree is the self-linking number of the initial curve, and the orientation of the manifold prescribes how to compute it from a diagram.
Date received: November 11, 2008
Bar-Natan modules and Bar-Natan pairings of oriented 3-manifolds
by
Uwe Kaiser
Boise State University
We discuss sesquilinear pairings defined by Bar-Natan modules (and their generalizations using general Frobenius algebras), which descend from universal manifold pairings recently discussed by Calegari, Freedman, Walker and others. Such a Bar-Natan pairing exists for each oriented closed surface with an embedded oriented closed 1-manifold (and each Frobenius algebra with involution). We also discuss how the Heegaard genus of closed 3-manifolds naturally appears in the calculation of Bar-Natan modules, and more generally how the calculation of Bar-Natan modules is related with the geometric topology of the 3-manifold.
Date received: November 28, 2008
On an inequality between unknotting number and crossing number of links
by
Junsuke Kanadome
Graduate School of Education, Waseda University
Coauthors: Ryo Hanaki
It is well known that for any knot K, twice the unknotting number of K is less than or equal to the crossing number of K minus one and for any link L, twice the unknotting number of L is less than or equal to the crossing number of L. Taniyama characterized the knots and links that satisfy the equalities. We characterize the links where twice the unknotting number is equal to the crossing number minus one.
Date received: November 30, 2008
A Graphical Bracket Polynomial for Virtual Knots
by
Louis H. Kauffman
Univeristy of Illinois at Chicago
The invariant of virtual knots discussed in this talk is a relative of the Arrow Polynomial and Extended Bracket Polynomials discussed in arXiv:0712.2546 and arXiv:0810.3858. We analyze the oriented state expansion of the bracket polynomial and find that for knots in thickened surfaces and for virtual knots (and links) there is a stronger invariant obtained by a system of state reduction that retains much of the oriented structure. The graphical bracket G[K] takes values in virtual graphs with coefficient polynonmials, and will be used to determine the minimal genus of various examples. Progress in categorification will be discussed.
Paper reference: arXiv:0712.2546, arXiv:0810.3858
Date received: November 5, 2008
Braids and Open Book Decompositions
by
Keiko Kawamuro
Rice University, The Institute for Advanced Study
Coauthors: Elena Pavelescu
We construct an immersed surface for a braid in an annulus open book decomposition, which is a generalization of the Bennequin surface for a braid in R3. By resolving the singularities of the immersed surface, we obtain an embedded Seifert surface for the braid. We find a self-linking number formula associated to the surface and prove that it is a generalization of the Bennequin's self-linking formula for a braid in R3. We also prove that our self-linking formula is invariant up to mod k under transversal isotopy of the contact structure compatible with the open book decomposition.
Paper reference: arXiv:0901.0414
Date received: January 3, 2009
Diagrammatics of categorifications
by
Mikhail Khovanov
Columbia University
We review the role of two-dimensional diagrammatics in categorications of quantum algebras.
Paper reference: arXiv:0803.4121, arXiv:0804.2080, arXiv:0807.3250
Date received: November 3, 2008
Colored Turaev-Viro invariants of twist knots
by
Yuya Koda
Tokyo Institute of Technology
In 2007, Barrett, Garcia-Islas and Martins defined a new series of invariants, colored Turaev-Viro invariants, of a pair (M, L), where M is a closed oriented 3-manifolds and L is an oriented link embedded in M. These invariants are defined as state-sums on a special polyhedron, restricting only to states such that certain regions have a certain pre-fixed color. In this talk, we briefly review the definition of these invariants. Then we construct special polyhedrons for twist knots using (1, 1)-decomposition of them, and we provide a formula for colored Turaev-Viro invariants of twist knots using these spines.
Date received: November 29, 2008
A new twist on Lorenz links
by
Ilya Kofman
College of Staten Island, CUNY
Coauthors: Joan Birman
Lorenz knots are periodic orbits in the flow on R3 given by the Lorenz differential equations. We show that Lorenz links coincide with a natural generalization of twisted torus links, given by repeated positive twisting. Using this correspondence, we identify many of the simplest hyperbolic knots as Lorenz knots. We also show that both hyperbolic volume and the Mahler measure of Jones polynomials are bounded for infinite collections of hyperbolic Lorenz links.
Paper reference: arXiv:0707.4331
Date received: November 13, 2008
Categorification of Quantum Groups
by
Aaron Lauda
Columbia University
Crane and Frenkel proposed that 4-dimensional TQFTs could be obtained by categorifying quantum groups at root of unity using their canonical bases. In my talk I will explain how the quantum enveloping algebra of quantum sl(2) at generic q can be categorified using a diagrammatic calculus. If time permits I will also explain joint work with Mikhail Khovanov on how this construction can be generalized to quantum sl(n). No background on quantum groups will be assumed.
Paper reference: arXiv:0803.3652
Date received: November 4, 2008
The Khovanov width of closed 3-braids
by
Adam Lowrance
Louisiana State University
Khovanov homology is a bigraded homology theory that categorifies the Jones polynomial. The support of Khovanov homology lies on a finite number of slope 2 lines with respect to the bigrading. Khovanov width is a measure of how many such lines support Khovanov homology. In this talk, I will compute the Khovanov width of closed 3-braids. Also, the proof will be adapted to a similar theory called odd Khovanov homology.
Date received: December 5, 2008
Commensurability Classes of (-2, 3, n) pretzel knots
by
Melissa Macasieb
University of Maryland
Coauthors: Thomas Mattman
Let K be a hyperbolic (-2, 3, n) pretzel knot and M = S^3 - K its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knot complements in the commensurability class of M. Indeed, if n is not 7, we show that M is the unique knot complement in its class.
Date received: November 17, 2008
Problem of knot complement in lens spaces
by
Daniel Matignon
Université de Marseille-Provence
A knot k in a 3-manifold M is not determined by its complement if there exists a homeomorphism between its complement and the complement of another knot k', but no homeomorphism from (M, k) to (M, k'). The problem of the existence of such a knot was initially settled by Tietze in 1908 for M=S3, then solved by Gordon and Luecke in 1989; and later the problem was generalized for all compact and orientable 3-manifolds. The knot complement problem is closely relative to the existence of cosmetic pairs, i.e. a non-trivial Dehn surgery on k yields a 3-manifold homeomorphic to M.
This paper concerns the study of the cosmetic pairs when the M's are lens spaces. We consider separately the knots according to their geometric type : Seifert fibered knots, satellite knots or hyperbolic knots. For the two former cases, we explicitly give exhaustive and infinite families of such cosmetic pairs. Each pair in these families is a counter-example to the Gordon's conjecture, which asserts that the homeomorphims between M and Mk(r) are orientation reversing. In the latter case, We show that there is no cosmetic pair (M, k) when M is a lens space and k is a fibered knot.
Date received: November 11, 2008
A Diagramless Homology
by
Adam McDougall
University of Iowa
Link homologies are typically built from link diagrams as opposed to link equivalence classes. In this talk, a link homology will be built directly from the links themselves. For chain groups, we use certain surfaces which have boundary equal to the given link. Surface signature is used for the homological grading.
This diagramless homology seems to consist of copies of Khovanov homology and sometimes copies of a truncated version of Khovanov homology. For all links one can find an injection from the Khovanov homology for that link into the diagramless homology for the link.
Date received: November 27, 2008
The Quantum Hyperbolic Invariants of the Figure Eight Knot
by
Heather Molle
The University of Iowa
Baseilhac and Benedetti have created a quantum hyperbolic invariant for knots. They conjecture that, like the colored Jones polynomial, their invariant is related to the volume of the complement of the knot. I will show that, in the case of the figure eight knot, this is true for certain values of charges and flattenings.
Date received: November 12, 2008
Failing to disprove the smooth 4-d Poincare conjecture.
by
Scott Morrison
Microsoft Station Q
Coauthors: Michael Freedman, Robert Gompf, Kevin Walker
There have long been proposed counterexamples to the smooth 4-d Poincare conjecture, such as the Cappell-Shaneson spheres. Robert Gompf found particularly nice handle decompositions of these, which in particular have only 0-, 1-, 2- and 4-handles. This means that the meridianal linking circles to the 2-handles are slice in the Cappell-Shaneson 4-balls, and thus showing that these linking circles are not slice in the standard 4-ball would be enough to see that the Cappell-Shaneson spheres really are counterexamples! Khovanov homology, and in particular the s-invariant, provides an obstruction to sliceness in the standard 4-ball.
Unfortunately, the smallest example is really large -- a two-component link with more than 200 crossings! I'll describe our efforts to compute the s-invariant anyway, involving a little bit of new mathematics, the newest and fastest program for computing Khovanov homology, and some serious computing power. And I'll try not to spoil the surprise of the final answer...
Date received: November 10, 2008
Seifert surgeries on knots and their network
by
Kimihiko Motegi
Nihon University, Japan
Coauthors: Arnaud Deruelle, Katura Miyazaki
Let K be a hyperbolic knot in the 3-sphere having a Seifert surgery, i.e. a Dehn surgery yielding a Seifert fiber space. There are several ways to show that the resulting manifold is actually Seifert fibered. On the other hand, there was no way to explain "why" the hyperbolic knot K has such a Seifert surgery. To find a natural explanation to the production of Seifert surgeries, we have introduced the Seifert Surgery Network in which a vertex is a pair (K, m) of a knot K and an integer m such that the result of the m-surgery on K is a Seifert fiber space, where a Seifert fiber space may have a fiber of index zero as a degenerate fiber. The networking viewpoint enables us to draw a global picture of Seifert surgeries and clarify relationships among those Seifert surgeries. In particular, if we have a path from a Seifert surgery (K, m) on a hyperbolic knot K to a Seifert surgery (Tp, q, n) on a torus knot Tp, q, then we can regard (Tp, q, n) as an "origin" of (K, m) and the path explains the production of (K, m). In this talk, we look at some particular examples and then present a list of Seifert surgeries on torus knots which are "origins" of those on hyperbolic knots. We will also discuss some related results.
Date received: October 31, 2008
On the number of irreducible components of a slice of the character variety of a knot group
by
Fumikazu Nagasato
Department of Mathematics, Meijo University
I will talk about a relationship between a (topological) property of a knot K and the number of irreducible components of a certain slice S0(K) of the SL2(C)-character variety of a knot group G(K).
Date received: November 27, 2008
Known and new results on intrinsically knotted and linked graphs
by
Ramin Naimi
Occidental College, Los Angeles, CA 90041
Coauthors: Erica Flapan, Blake Mellor
A graph is called intrinsically knotted (linked) if every embedding of it in S3 contains a nontrivial knot (link). In the early 1980's, Sachs, and independently, Conway and Gordon, showed that K6, the complete graph on six vertices, is intrinsically linked. Conway and Gordon also showed that K7 is intrinsically knotted. In 1995 Robertson, Seymour, and Thomas classified all intrinsically linked graphs by proving a conjecture of Sachs: a graph is intrinsically linked if and only if it contains as a minor either K6 or a graph obtained from K6 by triangle-Y and Y-triangle moves. We will start with the Conway and Gordon theorems and then go through a (partial) survey of old and recent results including this theorem: Let m be any positive integer; then for all sufficiently large n every embedding of Kn contains an m-component link such that any two components of the link have linking number at least m and the second coefficient a2 of the Conway polynomial of every component is at least m.
Paper reference: math/0610501
Date received: October 7, 2008
Cn-moves and periodic knots
by
Takuji Nakamura
Osaka Electro-Communication university
For each local move T, we can define the T-gordian distance for two knots. Our motivation is to research a relationship between the periodicity of knots and T-gordian distance. In this talk, we pay attention to Cn-move. In fact we show that for any p and any n( > 2) there exists a periodic knot of period p whose Cn-gordian distance to the trivial knot is one. We will mention this result about other local moves.
Date received: November 27, 2008
Link invariants from finite racks
by
Sam Nelson
Claremont McKenna College
We construct invariants of unframed knots and links from finite racks and show how these invariants extend the familiar quandle counting invariants. We discuss enhancements of these invariants via rack 2-cocycles and rack polynomials.
Paper reference: 0808.0029
Date received: October 21, 2008
On a local move for virtual knots and links
by
Toshiyuki Oikawa
Tokyo Woman's Christian University, Japan
We define a local move called a CF-move on virtual link diagrams, and show that any virtual knot can be deformed into a trivial knot by using generalized Reidemeister moves and CF-moves. Moreover, we define a new virtual link invariant n(L) for a virtual 2-component link L whose virtual linking number is an integer. Then we give necessary and sufficient conditions for two virtual 2-component links to be deformed into each other by using generalized Reidemeister moves and CF-moves in terms of a virtual linking number and n(L).
Date received: November 29, 2008
Surface-links whose triple point numbers are exactly 2n
by
Kanako Oshiro
Hiroshima University
By a 2-component surface-link such that each component is non-orientable, S. Satoh proved that for any positive integer n, there exists a 2-component surface-link whose triple point number is exactly 2n. In this talk, we show some other examples of the theorem. One of them implies that for any positive integer n, there exists 2-component surface-link such that it is composed of an orientable surface and a non-orientable surface and the triple point number is exactly 2n.
Date received: December 1, 2008
Odd homology of tangles and cobordisms
by
Krzysztof Putyra
Jagiellonian University
On the previous conference I introduced cobordisms with chronology, a special projection on the unit interval [0, 1], and used them to build a chain complex for a given link diagram, similar to the one of Bar-Natan. The complex is a link invariant unpo chain-homotopies and slightly modified S/T/4Tu relations. Then, using an appropriate functor into the category of modules one can obtain computable invariants. I will show that for modules over an integral domain, only two theories can be obtained in this way: Khovanov's and the odd one.
Another think is to define the chain complex for tangles, using cobordisms with corners. After a small refinement of this category, one can extend the construction to tangle cobordisms as well. This can lead us to invariants of knotted surfaces, however defined only upto sign.
Date received: December 17, 2008
Palindrome presentations of rational knots
by
Dušan Repovš
University of Ljubljana, Slovenia
We give special presentations of the groups of rational knots whose relators are palindromes. This means that the relators read the same forwards and backwards as words in the generators. Using such presentations, we obtain simple alternative proofs of some classical results concerning the Alexander polynomial and the character variety of rational knots. This is joint work with A. Cavicchioli and F. Spaggiari (to appear in J. Knot Theory).
Date received: October 10, 2008
Frobenius modules and essential surface cobordisms
by
Masahico Saito
University of South Florida
Coauthors: J. Scott Carter
A formulation of an algebraic structure is proposed, that describes essential surface cobordisms in 3-manifolds. It is a refinement of (1+1)-TQFTs, and has module and comodule structures over a Frobenius algebra with additional conditions. This structure gives a unified algebraic view of the differentials of a generalization of Khovanov homology defined by Asaeda-Przytycki-Sikora for thickened surfaces and those defined by Ishii-Tanaka for virtual knots. Constructions of new examples are also discussed.
Date received: November 30, 2008
Plane diagrammatics and categorification
by
Radmila Sazdanovic
The George Washington University
Coauthors: Mikhail Khovanov
We consider different categorifications of the ring of one variable polynomials which lead to caegorifications of some special functions.
Date received: November 27, 2008
Spatial graph diagrams with prescribed subdiagram partitions and their applications.
by
Reiko Shinjo
Osaka City University Advanced Mathematical Institute
We give a natural extension of the result given by J. H. Lee and G. T. Jin for link diagrams to spatial graph diagrams. As an application, we show some results for complementary regions of spatial graph projections.
Date received: December 1, 2008
Open string topology and knots
by
Michael Sullivan
University of Massachusetts, Amherst
I describe some new string topology operations and apply them to knots.
Date received: December 10, 2008
Khovanov homology for virtual links with two types of maps for Möbius cobordisms
by
Kokoro Tanaka
Tokyo Gakugei University
Coauthors: Atsushi Ishii (University of Tsukuba)
Khovanov homology is a homology theory for (classical) links which is a categorification of the Jones polynomial. If we want to extend Khovanov homology to virtual links, Khovanov's construction does not immediately work and the main difficulty arising is the existence of Möbius cobordisms (bifurcations of type 1 → 1). Recently, V. O. Manturov succeeded in extending Khovanov homology to virtual links. In his construction, the zero map is assigned to each of the Möbius cobordisms because of the grading reasons.
In this talk, we construct a new extension of Khovanov homology to virtual links by taking suitable grading shifts derived from the Miyazawa polynomial, which is known as a multi-variable generalization of the Jones-Kauffman polynomial. In our construction, we assign one of two non-zero maps to each of the Möbius cobordisms.
Date received: December 1, 2008
Knots yielding homeomorphic lens spaces by Dehn surgery
by
Masakazu Teragaito
Hiroshima University, Japan
Coauthors: Toshio Saito (Nara Women's University)
We show that there exist infinitely many pairs of distinct knots in the 3-sphere such that each pair can yield homeomorphic lens spaces by the same Dehn surgery. Moreover, each knot of the pair can be chosen to be a torus knot, a satellite knot or a hyperbolic knot, except that both cannot be satellite knots simultaneously. This exception is shown to be unavoidable by the classical theory of binary quadratic forms.
Paper reference: arXiv:0808.2852
Date received: November 5, 2008
Delta-cobordism of certain satellite links
by
Tatsuya Tsukamoto
Osaka Institute of Technology
Coauthors: Tetsuo Shibuya, Akira Yasuhara
Delta-cobordism is the equivalence relation generated by cobordism and self delta-moves. We study a relation between delta-cobordism of certain satellite links and delta-cobordism of their cores.
Date received: November 22, 2008
Presheaves, posets and Khovanov homology
by
Paul Turner
Fribourg/Heriot-Watt
I will outline some of the homology theory of presheaves of modules over posets and discuss how this is relevant to constructions in Khovanov homology. In particular I will discuss two applications of this point of view: (1) By fixing a number of crossings in a given link diagram one may build a cube of diagrams. I will explain how to construct a spectral sequence computing Khovanov homology for this situation; (2) It is known by the work of Przytycki that the Khovanov homology of a graph developed by Helme-Guizon-Rong, calculates Hochschild homology through a range of dimensions when the graph is the n-gon. I will explain how to extend this to define a homology theory for (possibly non-commutative) algebras starting with an arbitrary oriented graph.
Date received: November 14, 2008
Strangeness of immersed or real algebraic curves in the projective plane
by
Oleg Viro
Stony Brook University
The strangeness, Arnold's first degree invariants of generic immersions of the circle to the plane, is generalized to generic immersions of circle to the projective plane, and further, to generic real algebraic plane projective curves dividing complexification.
Date received: December 3, 2008
Grid Diagrams, Jones Polynomial and Khovanov Homology
by
Emmanuel Wagner
University of Aarhus
Coauthors: Jean-Marie Droz
Starting from a grid diagram of an oriented link, we will explain how to obtain the Jones polynomial of this link. We will also explore connections of this model with categorifications of the Jones polynomial.
Date received: November 25, 2008
A geometric description of colored HOMFLYPT homology
by
Ben Webster
MIT
Coauthors: Geordie Williamson
Building on previous work of the authors which gave a geometric description of Khovanov and Rozansky's HOMFLYPT homology, we give a construction of a categorification of the colored HOMFLYPT polynomial matching that of Mackaay, Stosic and Vaz. This construction is based on the equivariant cohomology of general linear groups and related spaces. While this description is more technically sophisticated than the bimodule approach introduced by Khovanov, it shows why the complexes of bimodules considered are natural choices and simplifies the proof of invariance.
Date received: November 10, 2008
Quasi-alternating Montesinos links
by
Tamara Widmer
University of Zürich
The aim of this talk will be to introduce new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, as provided bounds on unknotting number and slice genus, characterization of quasi-alternating links becomes an interesting open problem. We show that there exist classes of non-alternating Montesinos links, which are quasi-alternating.
Paper reference: arXiv:0811.0270
Date received: December 1, 2008
Riley Polynomials of 2-Bridge Knots
by
Susan Williams
University of South Alabama
Coauthors: Daniel Silver
Two-bridge knots are parameterized by pairs of relatively prime integers (α, β), 0 < β < α, with at most two pairs determining the same knot type k(α, β). R. Riley described a procedure for associating to each such pair a polynomial Φα, β with roots corresponding to the nonabelian parabolic SL2C representations of the knot group. (The representation is said to be parabolic if the image of any meridian has trace 2.)
We survey known results about this Riley polynomial, and present some new ones. In particular, if Φα, β is composite then at least one irreducible factor is not the Riley polynomial of a 2-bridge knot. If exactly one factor φ of Φα, β is not the Riley polynomial of a 2-bridge knot, and k(α, β) is not a torus knot, then the roots of φ correspond to faithful representations.
Date received: December 2, 2008
Classification of string links up to self delta-moves and concordance
by
Akira Yasuhara
Tokyo Gakugei University
For an n-component (string) link, the Milnor's concordance invariant is defined for each sequence I=i1i2...im (ij ∈ {1, ..., n}). Let r(I) denote the maximum number of times that any index appears in I. We show that two string links are equivalent up to self delta-moves and concordance if and only if their Milnor invariants with r ≤ 2 coincide.
Date received: November 19, 2008
Toward categorification of MOY link invariant via matrix factorization
by
Yasuyoshi Yonezawa
Nagoya University
Khovanov and Rozansky constructed a homological link invariant whose Euler characteristic is HOMFLY polynomial via matrix factorization. In my talk, we discuss a cagegorification of MOY polynomial (colored HOMFLY polynomial) via matrix factorization as a generalization of Khovanov-Rozansky homology.
Paper reference: arXiv:0806.4939
Date received: November 29, 2008
Copyright © 2008 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.