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Organizers |
Generalized Derivation Algebras and Applications
by
Amine Ben Abdeljelil
University of South Florida
Coauthors: Mohamed Elhamdadi, Abdenacer Makhlouf, Ivan Kaygodorov
In this talk, I will introduce some generalizations of a derivation of an algebra A. In particular, I will define the generalized derivations of n-ary BiHom Lie algebras.
Also, I will present some applications of the derivation algebras.
Date received: January 3, 2019
Torsion in the Khovanov homology of locally homologically thin links
by
Alex Chandler
North Carolina State University
Coauthors: Adam Lowrance, Radmila Sazdanovic, Victor Summers
We give a local version of Shumakovitch's result stating that homologically thin links have only torsion of order 2 in Khovanov homology. As an application, we show that an infinite family of 3-strand braids, strictly containing the 3-strand torus links, have only torsion of order 2 in Khovanov homology, thus giving a partial answer to the conjecture of Przytycki and Sazdanovic that 3-braids have only torsion of order 2. This allows us to give explicit computations of the integral Khovanov homology for all links in this infinite family.
Date received: January 8, 2019
The Jones Polynomial of Quasi-alternating Links
by
Nafaa Chbili
UAEU
Coauthors: Khaled Qazaqzeh (Kuwait University)
The aim of this talk is to introduce obstruction criteria to links' quasi-alternatness obtained in terms of their polynomial invariants. In particular, we discuss a conjecture about the behavior of the Jones polynomial of quasi-alternating links.
Date received: October 18, 2018
Conjectures on the Relations of Linking and Causality in Causally Simple Spacetimes
by
Vladimir Chernov
Dartmouth College
We formulate the generalization of the Legendrian Low conjecture of Natario and Tod (proved by Nemirovski and myself before) to the case of causally simple spacetimes. We prove a weakened version of the corresponding statement.
In all known examples, a causally simple spacetime (X, g) can be conformally embedded as an open subset into some globally hyperbolic (X̃, g̃) and the space of light rays in (X, g) is closely related to an open submanifold of the space of light rays in (X̃, g̃). If this is always the case, this provides an approach to solving the conjectures relating causality and linking in causally simple spacetimes.
Date received: November 14, 2018
A spectral sequence from Khovanov homology to knot Floer homology
by
Nathan P. Dowlin
Dartmouth College
Khovanov homology and knot Floer homology are two knot invariants which are defined using very different techniques, with Khovanov homology having its roots in representation theory and knot Floer homology in symplectic geometry. However, they seem to contain a lot of the same topological data about knots. Rasmussen conjectured that this similarity stems from a spectral sequence from Khovanov homology to knot Floer homology. In this talk I will describe a construction of this spectral sequence. The construction utilizes a recently defined knot homology theory HFK_2 which provides a framework in which the two theories can be related.
Date received: January 15, 2019
Transverse-Legendrian knots
by
Ivan Dynnikov
Steklov Mathematical Institute of Russian Academy of Sciences
By a transverse-Legendrian knot I call a knot which is transverse with respect to the standard contact structure on the three-sphere and Legendrian with respect to the mirror image of the standard contact structure. There is a natural combinatorial way to represent such knots so that each isotopy class admits only finitely many presentations, which can be easily searched. So the problem of comparing transverse-Legendrian knots viewed up to isotopy (within the class of transverse-Legendrian knots) has a simple algorithmic solution.
Combined with our recent joint results with M.Prasolov and V.Shastin, this allows one to distinguish transverse knots in many cases when the known transverse knot invariants fail to do so.
Date received: December 18, 2018
Ring Theoretic Aspects of Quandles.
by
Mohamed Elhamdadi
University of South Florida
Coauthors: N. Fernando and B. Tsvelikhovskiy.
In a recent paper arXiv:1709.03069 Bardakov et al. initiated the study of quandle rings and posed some open problems. We will start the talk by giving some basic properties of these rings, we will explain how one obtains a complete description of right ideals under the assumption that the inner automorphism group Inn(X) acts orbit 2-transitively on X. We will show that if for two quandles X and Y the inner automorphism groups act 2-transitively and k[X] is isomorphic to k[Y ], then the quandles are of the same partition type. We will also give explicit examples when the quandle rings k[X] and k[Y ] are isomorphic, but the quandles X and Y are not isomorphic, thus solving some open problems in the paper by Bardakov et al.
Date received: November 14, 2018
Singular Based Matrices for Virtual 2-Strings
by
David Freund
Harvard University
A singular virtual 2-string α is a wedge of two circles on a closed oriented surface. Up to equivalence by virtual homotopy, α can be realized on a canonical surface Σα. We use the homological intersection pairing on Σα to associate an algebraic object to α called a singular based matrix. In this talk, we show that these objects can be used to distinguish virtual homotopy classes of 2-strings and to compute the virtual Andersen-Mattes-Reshetikhin bracket of families of 2-strings.
Date received: January 4, 2019
Moduli of Tangle Invariants
by
Charles Frohman
The University of Iowa
It is well known that given a semisimple lie algebra there is a one parameter family of tangle invariants based on the representation theory of the associated quantum group. An understanding of the invariant theory of the quantum group leads to an invariant of tangles based on framed colored graphs. We turn the process on its head. Given a system of colored webs and linear relations between them, what are the invariants of framed links based on them? We get moduli spaces of link polynomials. The components of the moduli space correspond to the representation theory of different algebras.
In this lecture we will describe the theoretical setup for the theory. We will then describe some preliminary results of the author and his students.
Date received: October 10, 2018
Künneth formulae in persistent homology
by
Hitesh Gakhar
Michigan State University
Coauthors: Jose A. Perea
In topology, Künneth theorem gives a relationship between the homology of the product space and that of the factor spaces. We show similar theorems for persistent homology. That is, for two different notions of product filtrations, we give relationships between the persistent homology of the product filtration(s) and that of the factor filtered spaces.
Date received: November 11, 2018
Describing surfaces and isotopies in 4-manifolds via banded unlinks
by
Mark Hughes
Brigham Young University
Coauthors: Seungwon Kim and Maggie Miller
There are a number of well-established ways to represent knotted surfaces and isotopies between them in S^4, including motion pictures with movie moves, or broken surface diagrams with Roseman moves. In this talk I will discuss another method of representing surfaces in 4-space via banded unlink diagrams, which can also be used to describe surfaces in an arbitrary oriented 4-manifold X. I will present a set of moves which are sufficient to relate any two banded unlink presentations of isotopic surfaces in X, which generalizes a theorem in S^4 due to Swenton. Time permitting I will outline some applications of this approach to studying surfaces via banded unlinks.
Date received: December 10, 2018
Quandle invariants via bridge trisections
by
Jason Joseph
University of Georgia
Quandle cocycle invariants have been used by Carter, Jelsovsky, Kamada, Langford, Saito, Satoh, and others to detect properties of surface knots such as triple point number, reversibility, and ribbon concordance. In 2016 Meier and Zupan introduced bridge trisections, a new way to look at knotted surfaces. Here the data of the surface is expressed by three 1-dimensional tangles. In this talk I will show how to compute the quandle 3-cocycle and 2-cocycle surface invariants directly from the tangles of a bridge trisection diagram. In the process, I will show how to obtain a broken surface diagram from a bridge trisection diagram and how to compute the peripheral subgroup.
Date received: November 28, 2018
Virtual Knot Cobordism
by
Louis H Kauffman
University of Illinois at Chicasgo
This talk will discuss problems in virtual knot cobordism and properties of invariants such as the
concordance invariance of the Affine Index Polynomial and the infinity of pass classes of virtual knots.
Date received: November 13, 2018
Braid index of knotted surfaces
by
Sudipta Kolay
Georgia Tech
Braided surfaces play a similar role in understanding the knotted surfaces in R^4, as classical braids for knots in R^3. In this talk, I will introduce the notion of braided surfaces, and discuss properties of braid index of knotted surfaces and contrast it with that of classical knots.
Date received: December 21, 2018
Universal links with trivial Lagrangian
by
Slava Krushkal
University of Virginia
Analyzing the effect of Nielsen moves on good boundary links, we construct a collection of links, universal for 4-dimensional surgery, admitting Seifert surfaces with trivial Lagrangian. I will also discuss a condition on good boundary links, giving rise to a new class of slice links. (Joint work with Michael Freedman)
Date received: January 7, 2019
fully augmented links in the thickened torus
by
Alice Kwon
CUNY Graduate Center
I will be talking about augmented and fully augmented links in the thickened torus. More specifically, I will be discussing a decomposition of certain augmented links into ideal tetrahedra with angle structures to show that these augmented links in the thickened torus are hyperbolic. For fully augmented links I show they have a decomposition into what are called ideal torihedra with orthogonal dihedral angles. The orthogonal dihedral angles induce a complete hyperbolic structure for fully augmented links in the thickened torus.
Date received: December 23, 2018
An Implementation of Dynamic Programming Algorithm for Computing the coefficients of the Homflypt Polynomial
by
David Ledvinka
The University of Toronto
We present an implementation of the algorithm from the paper "The first coefficient of Homflypt and Kauffman polynomials: Vertigan proof of polynomial complexity using dynamic programming" by J. H. Przytycki, which allows one to compute the first several coefficients of the Homflypt polynomial of a knot in polynomial time. For example we were able to compute the first coefficient of a (randomly generated) 1000 crossing knot in under a minute.
Date received: November 11, 2018
Dihedral Knot Projections and Their Associated Knots and Links
by
Paul Lopata
Laboratory for Physical Sciences
I introduce a two-parameter family of geometric figures that serve well as knot projections. These figures, referred to as dihedral knot projections (on account of their symmetries), are used to generate knot and link diagrams in a straightforward manner. In this talk, I describe these dihedral knot projections and their associated knots and links, and discuss some of their more interesting properties. Of particular note are the alternating knots formed in this way. I demonstrate that, given any two non-trivial alternating knots generated using this method, these two knots are distinct if and only if their corresponding dihedral knot projections are distinct (up to chirality).
Date received: December 17, 2018
On the question of genericity of hyperbolic knots and links
by
Andrei Malyutin
Steklov Inst. St.Petersburg
A well-known conjecture in knot theory says that the proportion of hyperbolic knots among all of the prime knots of n or fewer crossings approaches 1 as n approaches infinity. We show that this conjecture contradicts several other plausible conjectures, including the 120-year-old conjecture on additivity of the crossing number of knots under connected sum and the conjecture that the crossing number of a satellite knot is not less than that of its companion. Also, we show that the proportion of hyperbolic links among all of the prime links of n or fewer crossings does not tend to 1 as n approaches infinity.
Date received: January 2, 2019
Jones Polynomial, Khovanov Homology and Higher Twist numbers for Weave knots
by
Rama Mishra
IISER Pune, India
Coauthors: Ross Staffeldt
We compute the Jones polynomial for the family W(3, n) , Weave Knots of type (3, n). We also determine the ranks of their Khovanov Homology and using the coefficients of Jones polynomial we estimate the Volume and the Higher Twist numbers for this class of knots.
Date received: September 28, 2018
On the tunnel number and the cutting number of handlebody-knots
by
Tomo Murao
University of Tsukuba
For a handlebody-knot H, a constituent handlebody-knot of H is a handlebody-knot obtained from H by removing an open regular neighborhood of some meridian disks of H. In this talk, we provide necessary conditions to be constituent handlebody-knots by using G-family of quandles colorings. Furthermore, as the corollaries, we give lower bounds for the tunnel number and the cutting number of handlebody-knots, which are dual geometric invariants for handlebody-knots.
Date received: December 30, 2018
gl(1|1) and an odd annular Bar-Natan category
by
Casey Necheles
Syracuse University
Coauthors: Stephan Wehrli
We introduce an odd version of the annular Bar-Natan category and show it is equivalent to a dotted version of the odd Temperley-Lieb supercategory defined by Brundan and Ellis. After setting dots equal to zero, this category embeds into the representation category of the Lie superalgebra gl(1|1). This gives us an interpretation of the gl(1|1) action on odd annular Khovanov homology shown in a recent paper by Grigsby and Wehrli.
Date received: January 7, 2019
Quandle Coloring Quivers
by
Sam Nelson
Claremont McKenna College
Coauthors: Karina Cho (Harvey Mudd College)
We define a quiver-valued invariant of oriented knots and link associated to a finite quandle and derive from it several new computable link invariants.
Date received: October 10, 2018
Markov trace on cubic Hecke algebra
by
Stepan Orevkov
IMT, Universite Paul Sabatier, Toulouse France
Funar algebra K∞=K∞(α, β;k) is the quotient of the group algebra over a ring k of the braid group B∞ by two cubic relations: σ13−ασ12+βσ1−1=0 and another one which involves σ1 and σ2. The universal Markov trace on K∞ is the quotient map t of K∞(α, β, k[u, v]) to its quotient (as a k[u, v]-module) by trace relations xy=yx and by Markov relations σnx=ux, σn−1x=vx for x ∈ Kn. It is easy to check (due to the specific form of Funar's relation) that the quotient is of the form k[u, v]/I for some ideal I (i. e. that the trace t is determined by t(1)). We give an algorithm to compute the ideal I and we present the result of computations in some special cases. We also discuss how to define invariants of tranverse links using one-sided Markov trace on cubic Hecke algebras.
Date received: December 15, 2018
Snail links and random straight links
by
Nicholas Owad
Okinawa Institute of Science and Technology
We introduce straight number and snail links and present some recent results about this new family of knots and links. Then we will use straight diagrams to create a random link model and talk about their expected volume.
Date received: October 12, 2018
Semi-canonical Seifert Surfaces in Simple Symmetric Knot Covers
by
Ken Perko
325 Old Army Road (One Perko Court), Scarsdale, New York
Working at the intersection of some elementary ideas in classical knot theory -- covering spaces, 3-colored diagrams, and Seifert surfaces -- we construct lifted surfaces that cobound the index 2 branch of a "simple" covering space (i.e., one for which meridians correspond to transpositions) and show how certain crossing reversals preserve, or predictably modify, its linking number with the sum of all the other branches. Note that Hilden and Montesinos proved long ago that every closed orientable 3-manifold is a simple 3-fold knot cover.
Date received: December 8, 2018
Links with Topologically Minimal Bridge Spheres
by
Puttipong Pongtanapaisan
University of Iowa
Coauthors: Daniel Rodman
David Bachman introduced the notion of a topologically minimal surface as a generalization of a strongly irreducible surface. These surfaces are useful and have been utilized by Bachman to solve Gordon's Conjecture and the Stabilization Conjecture. It is a conjecture that topologically minimal surfaces have all of the same properties as geometrically minimal surfaces. Many Heegaard surfaces have been verified to be topologically minimal, but less is known about topologically minimal bridge surfaces. In this talk, I will discuss joint work in progress with Daniel Rodman finding topologically minimal bridge spheres in link complements.
Date received: December 11, 2018
Topologies on sets of polynomial knots and the homotopy types of the respective spaces
by
Hitesh Raundal
Harish-Chandra Research Institute, Prayagraj (Allahabad), India
A polynomial knot is a smooth embedding R\hookrightarrowRn such that the component functions are real polynomials. Let Pn be the set of all polynomial knots in Rn, and let P=∪n ∈ Z+Pn. By identifying a polynomial knot φ:R\hookrightarrowRn\hookrightarrowR∞ with a Λ-tuple (φij)i, j (where Λ = Z+×N and φ(t)=∑j(φij)itj for t ∈ R), we can think of the sets Pn and P as subsets of RΛ, and thus they can be given the subspace topologies that inherit from the box and the product topologies of RΛ. We also have the topologies on Pn and P induced by the metrics dr (for r ≥ 1) and d∞ given by dr(φ, ψ)=(∑i, j|φij-ψij|r)1/r and d∞(φ, ψ)=supi, j | φij-ψij| for φ, ψ ∈ P. We show that Pn has the same homotopy type as Sn-1 and P is contractible, where the spaces have any of the topologies described above.
Date received: November 10, 2018
Elastic knots
by
Philipp Reiter
University of Georgia
Coauthors: Heiko von der Mosel; Henryk Gerlach; Sören Bartels
It is an interesting experiment to form a knot in a piece of springy wire, stick the endpoints together, and release the configuration. Can we predict the resulting shape?
In this talk we will present an outline on this question which goes back to the work of Joel Langer and David Singer in the 1980s and report on recent developments.
Considering an elementary model which only relies on the bending energy of the centerline of the wire, the answer should only depend on the respective knot class.
We face a free obstacle problem that involves techniques at the interface of geometric analysis, low-dimensional topology, modeling, numerical analysis, and nonlinear optimization.
Date received: November 17, 2018
On non-orientable surfaces in 4-manifolds
by
Rustam Sadykov
Kansas State University
Coauthors: Dave Auckly
We find conditions under which a non-orientable closed surface embedded into an orientable closed 4-manifold X can be represented by a connected sum of an embedded closed surface in X and an unknotted projective plane in a 4-sphere. This allows us to extend the Gabai 4-dimensional light bulb theorem and the Auckly-Kim-Melvin-Ruberman-Schwartz one is enough theorem to the case of non-orientable surfaces.
Date received: December 17, 2018
Skein for the Yang-Baxter Homology
by
Masahico Saito
University of South Florida
Coauthors: Mohamed Elhamdadi, Emanuele Zappala
Homology theories for the Yang-Baxter equation (YBE) have been developed and studied, with applications to knot invariants and deformation theories. We introduce a skein computation for a YBE homology for the R-matrix corresponding to the Jones polynomial. A homology for such a matrix was defined by Przytycki and Wang by normalizing the R-matrix appropriately. We modify the skein relation accordingly for this normalization. Diagrammatic computations of low dimensional homology groups are presented.
Date received: January 5, 2019
Link homology, bridge trisections, and invariants of knotted surfaces
by
Adam Saltz
University of Georgia
I will describe an invariant of knotted surfaces in S^4 obtained by applying link homology to Meier and Zupan's bridge trisections. This invariant takes values in Z/2Z and distinguishes the unknotted sphere from the spun (2,3)-torus knot. I'll finish with some more speculative connections to transverse links and links in other three-manifolds.
Date received: December 8, 2018
The Jones Representations are Sesquilinear
by
Nancy C. Scherich
University of California, Santa Barbara
The Jones representations of the braid groups are parameterized by Young Tableaux.
I will give an inductive argument using the combinatorial structure of the Young Tableaux
to show the Jones representations are sesquilinear (a generalized notion of unitary).
Date received: November 25, 2018
On top Khovanov homotopy type
by
Marithania Silvero
Universidad del País Vasco
In 2017 we introduced, in a joint work with González-Meneses and Manchón, a simplicial complex whose cohomology equals the extreme Khovanov homology of a given link diagram. Later, in a joint work with Józef Przytycki, we conjectured that its homotopy type, if not contractible, is equivalent to a wedge of spheres.
In this talk we explore the relation between that construction and the Khovanov homotopy type introduced by Lipshitz and Sarkar, showing that both constructions are stably homotopy equivalent at the extreme quantum grading.
This is a joint work with Federico Cantero.
Date received: January 6, 2019
Translation distance bounds for fibered 3-manifolds with boundary
by
AJ Stas
CUNY Graduate Center
Given a properly embedded essential surface S with non-zero slope in a fibered hyperbolic 3-manifold M, we show that the translation distance of the monodromy (as it acts on the arc and curve complex of the fiber) can be bounded above by |χ(S)|. We use this result to show that essential surfaces become more complex in covers of M. Furthermore, we show that an infinite family of fibered hyperbolic knots satisfies a conjecture of Schleimer.
Date received: December 23, 2018
New quantum invariants and perturbative invariants of genus 2 handlebody-knots
by
Abe Sukuse
Osaka City University Advanced Mathematical Institute
A handlebody-knot is an embedding of a handlebody in the 3-sphere S3. We define the new quantum Uq (g) invariant and the perturbative g invariant of genus 2 handlebody-knots. It is effectively easy to calculate the perturbative invariant g=sl2, which is a significantly stronger invariant than the previous conventional invariant. These quantum invariants of handlebody-knots, unlike the other invariants, can distinguish handlebody-knots whose complementary space is homeomorophic.
Date received: November 20, 2018
Yet another categorification of the Alexander polynomial
by
Oleg Viro
Stony Brook University
For a Turaev-Reshetikhin vertex state sum functor from the category of oriented colored tangles to representations of quantum gl(1|1), a categorification is constructed. Similar constructions will be proposed for Turaev-Reshetikhin functors based on other quantum groups.
Date received: January 1, 2019
Classification of string links up to 2n-moves and link-homotopy
by
Kodai Wada
Waseda University
Coauthors: Haruko A. Miyazawa (Tsuda University), Akira Yasuhara (Waseda University)
We establish a relation between 2n-moves and Milnor link-homotopy invariants for string links. We have that two string links are equivalent up to 2n-moves and link-homotopy if and only if their Milnor link-homotopy invariants are congruent modulo n.
Date received: October 31, 2018
A Philosophy of Knots: Symbolic and Diagrammatic Reasoning
by
L.S. Wang
McGill University
This paper outlines braid/knot groups as a topic of interest for philosophers of mathematics independently, and explores the problem of axiomatizing braids and knots as an under-addressed philosophically salient parallel to grounding foundations. Representational flexibility of braids produces further insight into the establishment of a more non-arbitrary connection between symbolic and diagrammatic systems as logical languages. I provide a sound knot logic for foundations and demonstrate advantages on existing diagrammatic systems, with attention to some historical predecessors.
Date received: December 10, 2018
Colored quantum annular Khovanov homology
by
Stephan Wehrli
Syracuse University
Coauthors: A. Beliakova, M. Hogancamp, and K. Putyra
I will explain how to quantize the annular version of Khovanov homology that was defined by Asaeda-Przytycki-Sikora. The resulting quantized theory is strictly functorial and carries an action of quantum sl(2). Moreover, in the quantized annular setting, the Cooper-Krushkal complex categorifying the colored Jones polynomial becomes finite and homotopic to the nonreduced colored Khovanov complex.
Date received: December 3, 2018
Burnside groups and n-moves for links
by
Akira Yasuhara
Waseda University
Coauthors: Haruko A. Miyazawa (Tsuda University), Kodai Wada (Waseda University)
M. K. D abkowski and J. H. Przytycki introduced the nth Burnside group of a link, which is an invariant preserved by n-moves. Using this invariant, for an odd prime p, they showed that there exist links which cannot be reduced to trivial links via p-moves. It is generally difficult to determine if pth Burnside groups associated to a link and the corresponding trivial link are isomorphic. In this paper, we give a necessary condition for existence of such an isomorphism. Using this condition we give a simple proof for their results that concern p-move reducibility of links.
Date received: October 18, 2018
Higher order self-distributivity
by
Emanuele Zappala
University of South Florida
Coauthors: Mohamed Elhamdadi,
Masahico Saito
In this talk I will introduce the notion of higher order self-distributive operation, along with its geometrical interpretation and a (co)homology theory that generalizes the usual one for shelves.
I will present chain maps between binary rack complexes and certain suitably defined total complexes of ternary racks. I will also describe an "operadic" approach to the subject.
Date received: December 7, 2018
Copyright © 2018 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.