scholarly journals Quotients of the Gordian and H(2)-Gordian graphs

2021 ◽  
pp. 2150037
Author(s):  
Christopher Flippen ◽  
Allison H. Moore ◽  
Essak Seddiq
Keyword(s):  
Complete Graph ◽  
Jones Polynomial ◽  
Graph Isomorphism ◽  
Isomorphism Type ◽  
Equivalence Relations ◽  
Knot Invariants ◽  
Quotient Graphs ◽  
Gromov Hyperbolic ◽  
Vertex Sets

The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)-moves, respectively. We investigate quotients of these graphs under equivalence relations defined by several knot invariants including the determinant, the span of the Jones polynomial, and an invariant related to tricolorability. We show, in all cases considered, that the quotient graphs are Gromov hyperbolic. We then prove a collection of results about the graph isomorphism type of the quotient graphs. In particular, we find that the H(2)-Gordian graph of links modulo the relation induced by the span of the Jones polynomial is isomorphic with the complete graph on infinitely many vertices.

scholarly journals INTRODUCTION TO GRAPH-LINK THEORY

2009 ◽  
Vol 18 (06) ◽  
pp. 791-823 ◽  
Author(s):  
DENIS PETROVICH ILYUTKO ◽  
VASSILY OLEGOVICH MANTUROV
Keyword(s):  
Knot Theory ◽  
Jones Polynomial ◽  
Natural Generalization ◽  
Combinatorial Theory ◽  
Equivalence Relations ◽  
Virtual Link ◽  
Virtual Knot ◽  
Link Theory ◽  
Virtual Knot Theory ◽  
Graph Link

The present paper is an introduction to a combinatorial theory arising as a natural generalization of classical and virtual knot theory. There is a way to encode links by a class of "realizable" graphs. When passing to generic graphs with the same equivalence relations we get "graph-links". On one hand graph-links generalize the notion of virtual link, on the other hand they do not detect link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalization of the Kauffman–Murasugi–Thistlethwaite theorem on "minimal diagrams" for graph-links.

scholarly journals ON THE ANALYTIC PROPERTIES OF THE z-COLOURED JONES POLYNOMIAL

2005 ◽  
Vol 14 (04) ◽  
pp. 435-466 ◽  
Author(s):  
JOÃO FARIA MARTINS
Keyword(s):  
Jones Polynomial ◽  
Unitary Representations ◽  
Target Space ◽  
Analytic Extension ◽  
Torus Knots ◽  
Infinite Dimensional ◽  
Knot Invariants ◽  
Starting Point ◽  
Zero Radius ◽  
Formal Power

We analyse the possibility of defining ℂ-valued Knot invariants associated with infinite-dimensional unitary representations of SL(2,ℝ) and the Lorentz Group taking as starting point the Kontsevich integral and the notion of infinitesimal character. This yields a family of knot invariants whose target space is the set of formal power series in ℂ, which contained in the Melvin–Morton expansion of the coloured Jones polynomial. We verify that for some knots the series have zero radius of convergence and analyse the construction of functions of which this series are asymptotic expansions by means of Borel re-summation. Explicit calculations are done in the case of torus knots which realise an analytic extension of the values of the coloured Jones polynomial to complex spins. We present a partial answer in the general case.

scholarly journals Two Lectures on the Jones Polynomial and Khovanov Homology

2017 ◽  
Author(s):  
Edward Witten
Keyword(s):  
Gauge Theory ◽  
Three Dimensional ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Theory Approach ◽  
Laurent Polynomials ◽  
Four Dimensions ◽  
Knot Invariants ◽  
Fifth Dimension ◽  
Instanton Number

In the first of these two lectures I describe a gauge theory approach to understanding quantum knot invariants as Laurent polynomials in a complex variable q. The two main steps are to reinterpret three-dimensional Chern-Simons gauge theory in four dimensional terms and then to apply electric-magnetic duality. The variable q is associated to instanton number in the dual description in four dimensions. In the second lecture, I describe how Khovanov homology can emerge upon adding a fifth dimension.

MODULAR CATEGORIES AND 3-MANIFOLD INVARIANTS

1992 ◽  
Vol 06 (11n12) ◽  
pp. 1807-1824 ◽  
Author(s):  
VLADIMIR G. TURAEV
Keyword(s):  
Quantum Groups ◽  
Tensor Category ◽  
Jones Polynomial ◽  
Knot Invariants ◽  
Modular Tensor Category ◽  
Modular Category

The aim of this paper is to give a concise introduction to the theory of knot invariants and 3-manifold invariants which generalize the Jones polynomial and which may be considered as a mathematical version of the Witten invariants. Such a theory was introduced by N. Reshetikhin and the author on the ground of the theory of quantum groups. Here we use more general algebraic objects, specifically, ribbon and modular categories. Such categories in particular arise as the categories of representations of quantum groups. The notion of modular category, interesting in itself, is closely related to the notion of modular tensor category in the sense of G. Moore and N. Seiberg. For simplicity we restrict ourselves in this paper to the case of closed 3-manifolds.

scholarly journals Crosscap number of knots and volume bounds

2020 ◽  
Vol 31 (13) ◽  
pp. 2050111
Author(s):  
Noboru Ito ◽  
Yusuke Takimura
Keyword(s):  
Jones Polynomial ◽  
Upper Bounds ◽  
Crossing Number ◽  
Proved Theorem ◽  
Knot Invariants ◽  
Chord Diagrams ◽  
Hyperbolic Volume ◽  
Knot Invariant ◽  
Twist Number

In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for [Formula: see text] knots (Tables A.1).

scholarly journals Knot theory realizations in nematic colloids

2015 ◽  
Vol 112 (6) ◽  
pp. 1675-1680 ◽  
Author(s):  
Simon Čopar ◽  
Uroš Tkalec ◽  
Igor Muševič ◽  
Slobodan Žumer
Keyword(s):  
Phase Diagram ◽  
Knot Theory ◽  
Colloidal Particles ◽  
Jones Polynomial ◽  
Laser Tweezers ◽  
Knot Invariants ◽  
Topological Parameters ◽  
Twisted Nematic ◽  
2D Arrays ◽  
Knots And Links

Nematic braids are reconfigurable knots and links formed by the disclination loops that entangle colloidal particles dispersed in a nematic liquid crystal. We focus on entangled nematic disclinations in thin twisted nematic layers stabilized by 2D arrays of colloidal particles that can be controlled with laser tweezers. We take the experimentally assembled structures and demonstrate the correspondence of the knot invariants, constructed graphs, and surfaces associated with the disclination loop to the physically observable features specific to the geometry at hand. The nematic nature of the medium adds additional topological parameters to the conventional results of knot theory, which couple with the knot topology and introduce order into the phase diagram of possible structures. The crystalline order allows the simplified construction of the Jones polynomial and medial graphs, and the steps in the construction algorithm are mirrored in the physics of liquid crystals.

scholarly journals Toughness of $K_{a,t}$-Minor-free Graphs

10.37236/635 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Guantao Chen ◽  
Yoshimi Egawa ◽  
Ken-ichi Kawarabayashi ◽  
Bojan Mohar ◽  
Katsuhiro Ota
Keyword(s):  
Planar Graph ◽  
Positive Constant ◽  
Complete Graph ◽  
Planar Graphs ◽  
Spanning Tree ◽  
Maximum Degree ◽  
Free Graph ◽  
Minor Free Graphs ◽  
Vertex Sets ◽  
Connected Planar Graph

The toughness of a non-complete graph $G$ is the minimum value of $\frac{|S|}{\omega(G-S)}$ among all separating vertex sets $S\subset V(G)$, where $\omega(G-S)\ge 2$ is the number of components of $G-S$. It is well-known that every $3$-connected planar graph has toughness greater than $1/2$. Related to this property, every $3$-connected planar graph has many good substructures, such as a spanning tree with maximum degree three, a $2$-walk, etc. Realizing that 3-connected planar graphs are essentially the same as 3-connected $K_{3,3}$-minor-free graphs, we consider a generalization to $a$-connected $K_{a,t}$-minor-free graphs, where $3\le a\le t$. We prove that there exists a positive constant $h(a,t)$ such that every $a$-connected $K_{a,t}$-minor-free graph $G$ has toughness at least $h(a,t)$. For the case where $a=3$ and $t$ is odd, we obtain the best possible value for $h(3,t)$. As a corollary it is proved that every such graph of order $n$ contains a cycle of length $\Omega(\log_{h(a,t)} n)$.

On Some Equivalence Relations for Probabilistic Processes1

1992 ◽  
Vol 17 (3) ◽  
pp. 211-234
Author(s):  
Dung T. Huynh ◽  
Lu Tian
Keyword(s):  
Polynomial Time ◽  
Graph Isomorphism ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Isomorphism Problem ◽  
Equivalence Relations ◽  
Transition Systems ◽  
Bisimulation Equivalence ◽  
Finite Trace ◽  
Trace Equivalence

In this paper, we investigate several equivalence relations for probabilistic labeled transition systems: bisimulation equivalence, readiness equivalence, failure equivalence, trace equivalence, maximal trace equivalence and finite trace equivalence. We formally prove the inclusions (equalities) among these equivalences. We also show that readiness, failure, trace, maximum trace and finite trace equivalences for finite probabilistic labeled transition systems are decidable in polynomial time. This should be contrasted with the PSPACE completeness of the same equivalences for classical labeled transition systems. Moreover, we derive an efficient polynomial time algorithm for deciding bisimulation equivalence for finite probabilistic labeled transition systems. The special case of initiated probabilistic transition systems will be considered. We show that the isomorphism problem for finite initiated labeled probabilistic transition systems is NC(1) equivalent to graph isomorphism.

scholarly journals THE COMPLEXITY OF TOPOLOGICAL GROUP ISOMORPHISM

2018 ◽  
Vol 83 (3) ◽  
pp. 1190-1203 ◽  
Author(s):  
ALEXANDER S. KECHRIS ◽  
ANDRÉ NIES ◽  
KATRIN TENT
Keyword(s):  
Topological Group ◽  
Upper Bound ◽  
Graph Isomorphism ◽  
Equivalence Relations ◽  
Topological Isomorphism ◽  
Natural Numbers ◽  
Locally Compact ◽  
Group Isomorphism ◽  
Borel Reducibility ◽  
The One

AbstractWe study the complexity of the topological isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Borel spaces. For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound.

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