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Organizers |
Single photon self-interference as additive Curry-Howard correspondence
by
A Nait Abdallah
INRIA Paris Rocquencourt and UWO
This talk presents a logical analysis of single photon self-interference, using constructive logic and the logic of partial information, as well as a generalization of Curry-Howard isomorphism. The standard Curry-Howard correspondence, proofs-as-terms generalizes to an additive version, namely a "sets of proofs"-as-"formal sums of lambda-terms" correspondence. A "Feynman path-integral" paradigm can then be applied in this setting, provided lambda-terms are generalized to phase lambda-terms.
We discuss the application of this approach to a photon traversing a Mach-Zehnder interferometer, and some other interference problems.
Date received: November 25, 2012
Milnor Invariants and Gauss Diagram Formulas
by
Eleanor Abernthy
University of Tennessee, Knoxville
Coauthors: Stella Thistlethwaite
I will give a background of Gauss Diagram formulas and their relationship to Vassiliev Invariants. Gussarov proved that every Vassiliev Invariant has a Gauss Diagram formula (not unique) and Polyak and Viro have provided many examples of this. They also showed that the Milnor Invariant on three component links have a Gauss Diagram formula. I will briefly explain this background and then introduce my result of a Gauss Diagram formula for Milnor Invariants on almost trivial links with 4 or more components
Date received: November 16, 2012
Generalized crossing changes in satellite knots
by
Cheryl Balm
Michigan State University
We will show that if K is a satellite knot which admits a generalized cosmetic crossing change of order q with |q| ≥ 6, then K admits a pattern knot with a generalized cosmetic crossing change of the same order. As a consequence of this, we find that any prime satellite knot which admits a pattern knot that is fibered cannot admit a generalized cosmetic crossing change of order q with |q| ≥ 6. We also show that if there is any knot admitting a generalized cosmetic crossing change of order q with |q| ≥ 6, then there must be such a knot which is hyperbolic.
Date received: October 23, 2012
The Kakimizu complex of a link
by
Jessica Banks
CRM-ISM, Montreal
We give an introduction to the Kakimizu complex of a link, covering a number of (fairly) recent results. In particular we will see that the Kakimizu complex of a knot may be locally infinite, that the Alexander polynomial of an alternating link carries information about its Seifert surfaces, and that the Kakimizu complex of a special alternating link is understood.
Date received: November 26, 2012
Exceptional Surgery and Bridge Distance
by
Ryan Blair
University of Pennsylvania
Coauthors: Marion Campisi, Jesse Johnson, Scott A. Taylor, Maggy Tomova
We demonstrate a lower bound on the genus of an essential surface or Heegaard surface in a 3-manifold obtained by non-trivial surgery on a link in terms of the bridge distance of a bridge surface for the link. Consequently, knots with high distance bridge surfaces do not admit non-trivial non-hyperbolic surgeries or non-trivial cosmetic surgeries.
Date received: November 29, 2012
Foams and sl(n) tangle cohomology
by
Carmen Caprau
California State University, Fresno
We construct an integer cohomology theory for oriented tangles via a special type of dotted foams and 4-valent webs, which, for the case of links, yields a categorification of the quantum sl(n) link invariant (for n > 3). Our construction is closely related to a rank n Frobenius extension and its associated 2-dimensional TQFT with dots, and provides efficient computations of the resulting invariant for tangles.
Date received: October 27, 2012
Reidemeister/Roseman-type Moves to Embedded Foams in 4-dimensional Space
by
J. Scott Carter
University of South Alabama
The dual to a tetrahedron consists of a single vertex at which four edges and six faces are incident. Along each edge, three faces converge. A 2-foam is a compact topological space such that each point has a neighborhood homeomorphic to a neighborhood of that complex. Knotted foams in 4-dimensional space are to knotted surfaces, as knotted trivalent graphs are to classical knots. The diagram of a knotted foam consists of a generic projection into 3-space with crossing information indicated via a broken surface. In this paper, a finite set of moves to foams are presented that are analogous to the Reidemeister-type moves for knotted graphs. These moves include the Roseman moves for knotted surfaces. Given a pair of diagrams of isotopic knotted foams there is a finite sequence of moves taken from this set that, when applied to one diagram sequentially, produces the other diagram.
Date received: October 15, 2012
Classification of Rational Knots of Low Degree in the 3-Sphere
by
Shane D'Mello
Stony Brook University
The classification of rational knots in the projective space, up to rigid isotopy, is known for knots of degrees 5 and less. Here we will classify rational knots in the 3-sphere that are of degrees 6 and less. The 3-sphere will be treated as a subvariety of RP4.
Date received: November 21, 2012
Invariants of 4-moves
by
Mieczyslaw K. Dabkowski
University of Texas at Dallas
We give an overview of the Nakanishi 4-move conjecture (1979) and
related 4-move question of Kawauchi (e.g. is any link of
2 components reduced by 4-moves to the trivial link of two components or a
Hopf link?) We present a possible counterexample: the parallel 2-cable of the trefoil
knot (the only possible counterexample of 12 crossings).
Date received: December 11, 2012
Algorithmic Content and Structure in Effective Vector Spaces
by
Rumen Dimitrov
Western Illinois University
The two mid-levels of the lattice L(3,Z₂) of subspaces of a 3-dimensional vector space over Z₂ provide a natural model of the Fano plane. We will show that this structure also appears when studying sublattices of infinite dimensional effective vector spaces. In the framework introduced by Metakides and Nerode the study of effective vector spaces can be reduced to the study of a particular computable ℵ₀-dimensional vector space denoted V_{∞}. The vectors in V_{∞} are the finitely nonzero ω-sequences of elements of the underlying computable field F and the operations are pointwise. The space V_{∞} has dependence algorithm and so we can construct a computable basis I₀ of V_{∞}. The lattice of computably enumerable subspaces(c.e.) of V_{∞} is richer than the structure of the lattice of c.e. subsets of I₀. In fact both L(V_{∞}) and L^{∗}(V_{∞}) (which is L(V_{∞}) modulo finite dimension) are non-distributive modular lattices. We will show that if F=Z₂ then for every n there is a particular principal filter in L^{∗}(V_{∞}) which is isomorphic to L(n,Z₂).
Date received: November 13, 2012
TQFTs from Quasi-Hopf Algebras and Group Cocycles
by
Jenny George
The Ohio State University
The original Hennings TQFT is defined for quasitriangular Hopf algebras satisfying various nondegeneracy requirements. We extend this construction to quasitriangular quasi-Hopf algebras with related nondegeneracy conditions and prove that this new ``quasi-Hennings'' algorithm is well-defined and gives rise to TQFTs. The ultimate goal is to apply this construction to the Dijkgraaf-Pasquier-Roche twisted double of the group algebra, and then show that the resulting TQFT is equivalent to a more geometric one, described by Freed and Quinn.
Date received: November 9, 2012
On stable Khovanov homology of torus knots
by
Eugene Gorsky
Stony Brook University
Coauthors: Alexei Oblomkov, Jacob Rasmussen
A theorem of Stosic shows that the Khovanov homology of (n,m) torus knots stabilize at large m. The limiting homology are tightly related to the colored homology of the unknot. I will describe a simple conjectural model for the stable Khovanov homology using the Koszul homology of an explicit non-regular sequence of quadratic polynomials. This model reproduces available experimental data, including the torsion. The corresponding Poincare series turns out to be related to the Rogers-Ramanujan identity.
Date received: October 19, 2012
Projective Geometry and Quantum Logic
by
Tobias Hagge
U.T. Dallas
Quantum Logics were first constructed by Birkhoff and Von Neumann in an attempt to describe the structure of quantum states. Projective geometry offers answers to many questions related to decidability, axiomatization, and expressive power of such logics. We give an overview of the answers to some of these questions and the variety of queries which may be expressed as quantum logical propositional formulas, using projective geometry as a guide.
Date received: December 7, 2012
Alexander and Markov type theorems for link cobordisms
by
Mark Hughes
SUNY Stony Brook
For surfaces in 4-space we can generalize the notion of classical braid closures to define surface braid closures, which give closed surfaces in R4. Viro proved that any isotopy class of surface link in R4 can be realized as a surface braid closure. Furthermore Kamada gave a set of moves analogous to Markov moves which are sufficient to join any two isotopic surface braid closures through a sequence of surface braid closures. In this talk I will discuss a relative version of these ideas which involves braiding cobordisms in R3 ×[0, 1]. These braided link cobordisms admit particularly nice motion picture descriptions, where all but a finite number of stills are braid closures, while the singular stills each contain a single critical point. We will show that any cobordism in R3 ×[0, 1] between braid closures in R3 ×{0} and R3 ×{1} is isotopic relative its boundary to a braided link cobordism, and that any two braided link cobordisms which are isotopic in R3 ×[0, 1] can be joined by a sequence of Markov type moves. These constructions make use of Morton's diagram threading technique and Kamada's generalization of Markov's theorem for closed surface braids.
Date received: November 20, 2012
A state model for the SO(2n) Kauffman polynomial
by
Dionne Ibarra
California State University, Fresno
Coauthors: Carmen Caprau and David Heywood
Francois Jaeger presented the two-variable Kauffman polynomial of an unoriented link L as a weighted sum of HOMFLY-PT polynomials of oriented links associated with L. Murakami, Ohtsuki and Yamada (MOY) used planar graphs and a recursive evaluation of these graphs to construct a state model for the sl(n) link invariant (a one-variable specialization of the HOMFLY-PT polynomial). We apply the MOY framework to Jaeger's work, and construct a state summation model for the SO(2n) Kauffman polynomial.
Date received: October 27, 2012
Quasi-triviality of quandles for link-homotopy
by
Ayumu Inoue
Tokyo Institute of Technology
We introduce the notion of quasi-triviality of quandles and define homology of quasi-trivial quandles. Quandle cocycle invariants are invariant under link-homotopy if they are associated with 2-cocycles of quasi-trivial quandles. We thus obtain numerical link-homotopy invariants.
Date received: October 31, 2012
Khovanov homology and Kirby moves
by
Noboru Ito
Waseda University
In this talk, we discuss some observations of behaviors of Khovanov homology under Kirby moves. We also consider its application.
Date received: December 4, 2012
On a monoid associated to knotted surfaces in special form
by
Michal Jablonowski
University of Gdansk
We introduce monoid corresponding to knotted surfaces in four sphere and present over a dozen relations among its four type of generators. Then we wish to investigate an index associated to the monoid also being invariant for knotted surface. Using our relations we will briefly prove that there are exactly six types of surfaces with index less or equal to two, and there are infinitely many types of knotted surfaces with index equal to three. Also as a straightforward application we will show different proof of well known theorem about twist-spun knots.
Date received: November 21, 2012
The characteristic-2 Rasmussen invariant and mutation
by
Thomas Jaeger
Syracuse University
We show that a characteristic-2 version of Rasmussen's s-invariant, which is implicit in the work of Turner and has recently been shown by Lipshitz and Sarkar to be a concordance invariant, does not change under mutation.
Date received: October 31, 2012
The geometry of state surfaces and the colored Jones polynomials
by
Effie Kalfagianni
Michigan State University
Coauthors: David Futer (Temple University)
Jessica S. Purcell (Brigham Young University)
Under a diagrammatic hypothesis the boundary slope of certain incompressible surfaces in a knot complement is determined by the growth of the degree of the colored Jones polynomial of the knot. For hyperbolic knots, we show that the geometric type of these surfaces in the Thurston trichotomy is also determined by a coefficient of the colored Jones polynomial of the knot.
Date received: November 18, 2012
Branched coverings and braided manifolds of low dimensions
by
Seiichi Kamada
Hiroshima University
Coauthors: J. Scott Carter
We introduce the notion of a braided manifold, discuss how it relates with branched coverings and how we describe it. Especially, for the 2-dimensional case, we have a method of describing branched coverings of the 2-sphere by using graphics, called a permutation chart. It is an unoriented version of a chart description of 2-dimensional braids. We can generalize this to 3-dimensional case, and even more higher. This is an on-going project with J. Scott Carter.
Date received: October 24, 2012
On cyclic equivalence classes of nanowords and finite type invariants
by
Yuka Kotorii
Tokyo Institute of Technology
Turaev developed the theory of words based on the analogy with curves, knots, etc, called nanowords. We extend the definition of finite type invariants for stably homeomorphic classes of curves on closed oriented surface to that for cyclicequivalence classes of nanowords and construct the universal ones.
Date received: November 24, 2012
Hochschild, Chevalley-Eilenberg and quandle homologies are braided homologies
by
Victoria Lebed
Université Paris 7, IMJ
In this talk we present a homology theory for braided objects in monoidal categories. We give a construction in terms of quantum co-shuffles and then refine the differential structure to get a presimplicial one, weakly simplicial if the braided object is endowed with a compatible comultiplication. Explicit formulas will be given using the graphical calculus. Several remarkable features of the braided homologies will be discussed.
On the other hand, we interpret associative, Leibniz/Lie and quandle structures in terms of braidings. Braided homology theories for these ``structural'' braidings include familiar homologies for the corresponding structures.
Date received: October 9, 2012
Maps on quantum states
by
Chi-Kwong Li
Department of Mathematics, College of William and Mary
The talk will focus on functions on quantum states leaving invariant some important subsets (such as the set of separable states), functions (such as the eigenvalues of uncorrelated states), or properties. Some recent results will be surveyed, and open problems will be mentioned.
The talk is based on recent papers with A. Fosner, S. Friedland, J. Hou, K. He, Z. Huang, Y.T. Poon, N.K. Sze.
Date received: November 13, 2012
Non-associative structures and their computability theoretic complexity
by
Kai Maeda
George Washington University
Richters degree of a countable algebraic structure is a Turing degree theoretic measure of the complexity of its isomorphism class. It has been shown that some structures, such as abelian groups or partially ordered sets, have arbitrary Richters degrees, by showing that their isomorphism classes contain infinite anti-chains of structures with certain algebraic and computability theoretic properties. I will extend these results to non-associative structures such as quandles.
Date received: November 26, 2012
A Knotty Bucket List
by
Paul Melvin
Bryn Mawr College
We have all spent most of our professional lives thinking about knots. They arise in myriads of contexts: in the study of 3 and 4-dimensional manifolds, in representation theory, in theoretical physics, in molecular biology, and so forth. In this talk I will present several knotty problems that relate to questions in 3 and 4-dimensional topology, that are perhaps not so well known, and for which there is no general consensus as to how they will be resolved. I have thought about these problems at one time or another, generally long ago, but now they are just sitting on the back burner. The 35th session of Knots in Washington seems like an opportune time to bring them (back) to your attention, and to promote them onto my own personal "knotty bucket list".
Date received: November 17, 2012
How many hyperbolic 3-manifolds can have the same volume?
by
Christian Millichap
Temple University
The work of Jorgensen and Thurston shows that there is a finite number N(v) of orientable hyperbolic 3-manifolds with any given volume v. In this talk, we construct examples showing that the number of hyperbolic knot complements with a given volume v can grow at least factorially fast with v. A similar statement holds for closed hyperbolic 3-manifolds, obtained via Dehn surgery. Furthermore, we give explicit estimates for lower bounds of N(v) in terms of v for these examples. These results improve upon the work of Hodgson and Masai, which describes examples that grow exponentially fast with v. Our constructions rely on performing volume preserving mutations along Conway spheres and on the classification of Montesinos knots.
Date received: October 24, 2012
Quantum Enhancements
by
Sam Nelson
Claremont McKenna College
Coauthors: Veronica Rivera
We define customized quantum invariants of birack-labeled knots and links to obtain an enhancement of the counting invariant.
Date received: November 7, 2012
Knots, categories, and dynamics
by
Maciej Niebrzydowski
University of Louisiana at Lafayette
The subject of my talk is a generalization of theories involving diagrams and moves on the diagrams, tying them closely with the notions of category theory and transition systems. In a nutshell: we consider diagrams decorated by structures (objects of a given category), the structure determines which moves are possible on a diagram, and the moves change the structure in a controlled way. This creates dynamics. The following is our main example. Given a classical link diagram, suppose that the components are ordered in some way, and that the components with higher order are allowed to move over components with the lower order, but not the other way around. What is the set of diagrams one can obtain in such case? More generally, we can have a binary relation R on the set of arcs of the diagram, and impose the condition that an arc b can move over an arc a only if aR b. We will investigate the consequences of such a condition. In particular, it is necessary to make a choice regarding the status (with respect to R) of the arcs created by the Reidemeister moves. There are some options, and they lead to different kinds of links; the choice can be influenced by the desired applications. In some cases, there arises a phenomenon of irreversible Reidemeister moves which leads to a different definition of a knot (now viewed as a subcategory) and to replacing invariants by indicators (functors). Still, if we decide to use the binary relation in which all the arcs are related, we return to classical knot theory. We define homologies of binary relations and merge them with quandle homology to get a tool for answering some of the questions that arise.
Date received: October 10, 2012
State surfaces in knot complements
by
Jessica S Purcell
Brigham Young University
Coauthors: David Futer, Efstratia Kalfagianni
In the last decade, there have been several breakthroughs in hyperbolic geometry and 3-manifold topology. One goal in the research community is to use these new results to shed light on related problems, for example problems in knot and link theory. In this talk, we will describe some of our work in this direction. We determine geometric information on incompressible spanning surfaces in a large class of knot and link diagrams. Study of the geometric properties of these surfaces allows us to detect fibers and bound volumes, and relate them to certain coefficients of the Jones polynomial. This is joint with David Futer and Efstratia Kalfagianni.
Date received: November 18, 2012
Quantum Dots, Squids, Cavity QED - A unified approach vie Lie theory
by
Viswanath Ramakrishna
Dept of Mathematical Sciences, University of Texas at Dallas
Coauthors: Zhigang Zhang, Goong Chen
We present a unified approach to three distinct types of two qubit systems - namely quantum dots, cavity QED and SQUIDS. We show that the Hamiltonians underpinning these three systems are very nearly conjugate, and that these conjugations can be found explicitly. We were lead to these conjugations by invoking the theory of KAK (or Cartan) decompositions (specifically of Type I) of the Lie group SU(4). In fact, this can be viewed as a very nice didcatic illustration of the various aspects (especially the non-uniqueness) of KAK decompositions.
Date received: November 18, 2012
Quandle Cocycle Invariant for Knotted 3-Manifolds in 5-Space.
by
Witold Rosicki
University of Gdansk
Quandle is an algebraic structure for which Carter, Jeslovsky, Kamada, Langford, and Saito (1999, 2003) have build cohomology theory. They have found invariants of knots and knotted surfaces in 2-nd and 3-rd quandle cohomology. An analogous invariant can be defined for knotted 3-manifolds. We can prove its correctness using Roseman moves. J. Przytycki and I are currently developing an analogous invariant for n-manifolds in codimension 2.
Date received: November 18, 2012
Genus ranges of 4-regular rigid vertex graphs
by
Masahico Saito
University of South Florida
Coauthors: Dorothy Buck, Egor Dolzhenko,
Natasha Jonoska, Karin Valencia
The genus range of a graph is the set of values of genera over all surfaces into which the given graph is embedded cellularly, and we study the genus ranges of four-regular graphs with rigid vertices and a single transverse component. The genus ranges are shown to be sets of consecutive integers. Among consecutive integers satisfying the Euler characteristic formula, we investigate which sets can be, or cannot be, realized as genus ranges. Computer calculations are presented, and problems, conjectures are discussed.
Date received: November 2, 2012
The pallet graph of a Fox coloring
by
Shin Satoh
Kobe University
Coauthors: Takuji Nakamura and Yasutaka Nakanishi
We introduce the notion of a graph associated with a Fox p-coloring of a knot, and prove that any non-trivial p-coloring requires at least [log_2 p]+2 colors. Moreover, we also prove that this lower bound is best possible in the sense that there is a p-colorable virtual knot which attains the bound.
Date received: October 23, 2012
A New Spectral Sequence in Khovanov Homology
by
Cotton Seed
Princeton University
Coauthors: Joshua Batson
Khovanov homology is an invariant of links L in S^3 which categorifies the Jones polynomial. In this talk, I will describe a new spectral sequence in Khovanov homology, the link splitting spectral sequence. The spectral sequence starts at the Khovanov homology of L and converges to the Khovanov homology of the disjoint union of the components of L. As an application, building on results of Kronheimer-Mrowka and Hedden-Ni, I will prove that Khovanov homology detects the unlink.
Date received: October 26, 2012
Quantum Computation and Quantum Simulation Experiments with Trapped Ions
by
Crystal Senko
Joint Quantum Institute, University of Maryland
Coauthors: Rajibul Islam, Wes C. Campbell, Simcha Korenblit, Aaron Lee, Phil Richerme, Jacob Smith, Jonathan Mizrahi, Kale Johnson, Brian Neyenhuis, Susan Clark, David Hayes, David Hucul, Volkan Inlek, Taeyoung Choi, Shantanu Debnath, Caroline Figgatt, Andrew Manning, Chenglin Cao, Ken Wright, and Christopher Monroe
The experimental implementation of a large-scale quantum computer remains a major outstanding challenge. Several physical systems have been demonstrated to have the excellent isolation from environmental noise and the precise external control needed to perform quantum computations, and some of the most advanced results have been achieved using trapped atomic ions. I will give an overview of how trapped ions are used for quantum information processing and briefly discuss the current state of trapped ion quantum computing experiments. I will also discuss our research group's current efforts into performing quantum simulations with trapped ions.
Date received: December 3, 2012
String-Net Condensation and its Application to Quantum Computing
by
Matthew Titsworth
Department of Physics, University of Texas at Dallas
Coauthors: Tobias Hagge
Topological quantum computing(TQC) is a quantum computing model which makes use of the ground state degeneracy of topological phases(TPs) to address the decoherence problem. A large class of topological phases can be classified using the string-net condensation model and its generalizations. In this expository talk we will define string-net models, indicate their relation to topological phases, and discuss their relevance to quantum computing.
Date received: November 21, 2012
4-moves and the Dabkowski-Sahi invariant of knots
by
Robert Todd
University of Nebraska at Omaha
Coauthors: Susan Hermiller UNL
Mark Brittenham UNL
We study the 4-move invariant for links in the 3-sphere developed by Dabkowski and Sahi, which is defined as a quotient of the fundamental group of the link complement. We develop techniques for computing this invariant and show that for several classes of knots it is equal to the invariant for the unknot; therefore, in these cases the invariant cannot detect a counterexample to the 4-move conjecture.
Date received: September 17, 2012
Simple ribbon fusions and genera of links
by
Tatsuya Tsukamoto
Osaka Institute of Technology
Coauthors: Kengo Kishimoto, Tetsuo Shibuya
Gabai and Scharlemann showed the superadditivity of genera of knots under a band sum, and that equality holds if and only if the band sum is just a connected sum.
However when we extend the theorem to links, we have a problem for the equality.
In fact, there is a band sum of a link L and the trivial 2-component link which produces a link L' such that the genus of L' is equal to the genus of L, but L' is not ambient isotopic to L (This fusion is realized by a simple ribbon fusion, which we introduce in the talk). However Goldberg's 2nd genera of L is not equal to that of L'. In this talk, we talk about the superadditivity of Goldberg's v-th genera of links under simple ribbon fusions.
Date received: November 13, 2012
Exact volume of hyperbolic 2-bridge links
by
Anastasiia Tsvietkova
Louisiana State University
We will discuss the question of relating a diagram of a link to the geometry of its complement, and, in particular, to its exact hyperbolic volume. W. Thurston suggested a method for computing volume of hyperbolic 3-manifolds, based on a triangulation of the manifold. It was implemented by J. Weeks in the program SnapPea, which produces a decimal approximation as a result. For hyperbolic 2-bridge links, we give formulae that allow one to find the exact volume, i.e. to construct a polynomial and to find volume as an analytic function of one of its roots. The computation is performed directly from a reduced, alternating link diagram.
Date received: November 28, 2012
A symmetric group action on the Khovanov homology of cables
by
Stephan Wehrli
Syracuse University
Following up on a conjecture that I stated at an earlier KIW conference, I will describe an action of the symmetric group on the Khovanov homology of the n-cable of a knot. I will show that this action factors through the Temperley-Lieb algebra at q=1, and I will use this result to outline a relationship with Khovanov's categorification of the nonreduced n-colored Jones polynomial.
Date received: November 22, 2012
Symbolic dynamics in the arithmetic hierarchy
by
Sebastian Wyman
University of Florida
Coauthors: Douglas Cenzer
Recently, Cenzer, Dashti, and King showed that the subshifts arising from the symbolic dynamics of computable functions on the cantor space are exactly the decidable Π10 subshifts and those arising by avoiding a c.e. set of words are exactly the Π01 subshifts. We define conservatively approximable functions and use them to find functions whose symbolic dynamics give rise to exactly the Π01 subshifts. We also give conditions on a computable set of avoidable words which give rise to exactly the decidable subshifts.
Date received: November 27, 2012
Is a 1-twist spin of a knotted trivalent graph unknotted?
by
Seung Yeop Yang
University of South Alabama
Coauthors: S. Carter
The twist spin of a knotted trivalent graph is a foam that has simple closed branch lines that do not cross. Thus the foam is a surface with addition disks attached. Using an idea of Marumoto and Nakanishi, we show that there is a nontrivial foam obtained by a 1-twist spinning of a knotted theta-curve.
Date received: October 31, 2012
The (g, b)-decompositions of iterated torus knots
by
Alexander Zupan
University of Texas at Austin
For any knot K in the 3-sphere, and for each non-negative integer g, the knot K has a bridge number with respect to a genus g Heegaard surface. With respect to certain types of generic knots K, these genus g bridge numbers are determined by the classical bridge number of K. However, we will demonstrate that iterated torus knots can realize a wide array of unexpected genus g bridge numbers.
Date received: October 24, 2012
Copyright © 2012 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.