Polynomial invariants for virtual marked graphs and virtual surface-link theory I

2017 ◽  
Vol 26 (12) ◽  
pp. 1750081
Author(s):  
Sang Youl Lee
Keyword(s):  
Natural Extension ◽  
Polynomial Invariants ◽  
Kauffman Bracket ◽  
Virtual Links ◽  
Link Theory ◽  
Marked Graphs ◽  
Bracket Polynomial ◽  
The Ideal

In this paper, we introduce a notion of virtual marked graphs and their equivalence and then define polynomial invariants for virtual marked graphs using invariants for virtual links. We also formulate a way how to define the ideal coset invariants for virtual surface-links using the polynomial invariants for virtual marked graphs. Examining this theory with the Kauffman bracket polynomial, we establish a natural extension of the Kauffman bracket polynomial to virtual marked graphs and found the ideal coset invariant for virtual surface-links using the extended Kauffman bracket polynomial.

scholarly journals A BRACKET POLYNOMIAL FOR GRAPHS, III: VERTEX WEIGHTS

2011 ◽  
Vol 20 (03) ◽  
pp. 435-462 ◽  
Author(s):  
LORENZO TRALDI
Keyword(s):  
Link Diagram ◽  
Kauffman Bracket ◽  
Marked Graphs ◽  
Marked Graph ◽  
Bracket Polynomial

In earlier work the Kauffman bracket polynomial was extended to an invariant of marked graphs, i.e. looped graphs whose vertices have been partitioned into two classes (marked and not marked). The marked-graph bracket polynomial is readily modified to handle graphs with weighted vertices. We present formulas that simplify the computation of this weighted bracket for graphs that contain twin vertices or are constructed using graph composition, and we show that graph composition corresponds to the construction of a link diagram from tangles.

scholarly journals A BRACKET POLYNOMIAL FOR GRAPHS, IV: UNDIRECTED EULER CIRCUITS, GRAPH-LINKS AND MULTIPLY MARKED GRAPHS

2011 ◽  
Vol 20 (08) ◽  
pp. 1093-1128 ◽  
Author(s):  
LORENZO TRALDI
Keyword(s):  
Knot Theory ◽  
Euler System ◽  
Link Diagram ◽  
Euler Systems ◽  
Kauffman Bracket ◽  
Marked Graphs ◽  
Bracket Polynomial ◽  
The Universe

In earlier work we introduced the graph bracket polynomial of graphs with marked vertices, motivated by the fact that the Kauffman bracket of a link diagram D is determined by a looped, marked version of the interlacement graph associated to a directed Euler system of the universe graph of D. Here we extend the graph bracket to graphs whose vertices may carry different kinds of marks, and we show how multiply marked graphs encode interlacement with respect to arbitrary (undirected) Euler systems. The extended machinery brings together the earlier version and the graph-links of Ilyutko and Manturov [J. Knot Theory Ramifications18 (2009) 791–823]. The greater flexibility of the extended bracket also allows for a recursive description much simpler than that of the earlier version.

scholarly journals A BRACKET POLYNOMIAL FOR GRAPHS, II: LINKS, EULER CIRCUITS AND MARKED GRAPHS

2010 ◽  
Vol 19 (04) ◽  
pp. 547-586 ◽  
Author(s):  
LORENZO TRALDI
Keyword(s):  
Jones Polynomial ◽  
Positive Contribution ◽  
Geometric Information ◽  
Combinatorial Structures ◽  
Negative Contribution ◽  
Link Diagrams ◽  
Virtual Links ◽  
Marked Graphs ◽  
Bracket Polynomial ◽  
The Relationship

Let D be an oriented classical or virtual link diagram with directed universe [Formula: see text]. Let C denote a set of directed Euler circuits, one in each connected component of U. There is then an associated looped interlacement graph [Formula: see text] whose construction involves very little geometric information about the way D is drawn in the plane; consequently [Formula: see text] is different from other combinatorial structures associated with classical link diagrams, like the checkerboard graph, which can be difficult to extend to arbitrary virtual links. [Formula: see text] is determined by three things: the structure of [Formula: see text] as a 2-in, 2-out digraph, the distinction between crossings that make a positive contribution to the writhe and those that make a negative contribution, and the relationship between C and the directed circuits in [Formula: see text] arising from the link components; this relationship is indicated by marking the vertices where C does not follow the incident link component(s). We introduce a bracket polynomial for arbitrary marked graphs, defined using either a formula involving matrix nullities or a recursion involving the local complement and pivot operations; the marked-graph bracket of [Formula: see text] is the same as the Kauffman bracket of D. This provides a unified combinatorial description of the Jones polynomial that applies seamlessly to both classical and non-classical virtual links.

scholarly journals A bridge between Conway–Coxeter friezes and rational tangles through the Kauffman bracket polynomials

2019 ◽  
Vol 28 (14) ◽  
pp. 1950083 ◽  
Author(s):  
Takeyoshi Kogiso ◽  
Michihisa Wakui
Keyword(s):  
Kauffman Bracket ◽  
Rational Tangles ◽  
Bracket Polynomial ◽  
Bracket Polynomials ◽  
Zigzag Type

In this paper, we build a bridge between Conway–Coxeter friezes (CCFs) and rational tangles through the Kauffman bracket polynomials. One can compute a Kauffman bracket polynomial attached to rational links by using CCFs. As an application, one can give a complete invariant on CCFs of zigzag-type.

GENERALIZED INDEX POLYNOMIALS FOR VIRTUAL LINKS

2012 ◽  
Vol 21 (14) ◽  
pp. 1250128
Author(s):  
KYEONGHUI LEE ◽  
YOUNG HO IM
Keyword(s):  
Knot Theory ◽  
Recursive Method ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knots ◽  
Virtual Links

We construct some polynomial invariants for virtual links by the recursive method, which are different from the index polynomial invariant defined in [Y. H. Im, K. Lee and S. Y. Lee, Index polynomial invariant of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. We show that these polynomials can distinguish whether virtual knots can be invertible or not although the index polynomial cannot distinguish the invertibility of virtual knots.

ON COMPUTING KAUFFMAN BRACKET POLYNOMIAL OF MONTESINOS LINKS

2010 ◽  
Vol 19 (08) ◽  
pp. 1001-1023 ◽  
Author(s):  
XIAN'AN JIN ◽  
FUJI ZHANG
Keyword(s):  
Closed Form ◽  
Explicit Expression ◽  
Computer Algebra ◽  
Jones Polynomial ◽  
Kauffman Bracket ◽  
Jones Polynomials ◽  
Bracket Polynomial ◽  
Bracket Polynomials ◽  
Closed Form Formula

It is well known that Jones polynomial (hence, Kauffman bracket polynomial) of links is, in general, hard to compute. By now, Jones polynomials or Kauffman bracket polynomials of many link families have been computed, see [4, 7–11]. In recent years, the computer algebra (Maple) techniques were used to calculate link polynomials for various link families, see [7, 12–14]. In this paper, we try to design a maple program to calculate the explicit expression of the Kauffman bracket polynomial of Montesinos links. We first introduce a family of "ring of tangles" links, which includes Montesinos links as a special subfamily. Then, we provide a closed-form formula of Kauffman bracket polynomial for a "ring of tangles" link in terms of Kauffman bracket polynomials of the numerators and denominators of the tangles building the link. Finally, using this formula and known results on rational links, the Maple program is designed.

scholarly journals Introduction to disoriented knot theory

2018 ◽  
Vol 16 (1) ◽  
pp. 346-357
Author(s):  
İsmet Altıntaş
Keyword(s):  
Knot Theory ◽  
Jones Polynomial ◽  
Polynomial Invariants ◽  
Linking Number ◽  
Link Diagrams ◽  
New Ideas ◽  
Numerical Invariants ◽  
Bracket Polynomial ◽  
Knots And Links

AbstractThis paper is an introduction to disoriented knot theory, which is a generalization of the oriented knot and link diagrams and an exposition of new ideas and constructions, including the basic definitions and concepts such as disoriented knot, disoriented crossing and Reidemesiter moves for disoriented diagrams, numerical invariants such as the linking number and the complete writhe, the polynomial invariants such as the bracket polynomial, the Jones polynomial for the disoriented knots and links.

scholarly journals Graph polynomials and link invariants as positive type functions on Thompson’s group F

2019 ◽  
Vol 28 (02) ◽  
pp. 1950006 ◽  
Author(s):  
Valeriano Aiello ◽  
Roberto Conti
Keyword(s):  
Elementary Proof ◽  
Tutte Polynomial ◽  
Positive Type ◽  
Polynomial Invariants ◽  
Graph Polynomials ◽  
Link Invariants ◽  
Kauffman Bracket ◽  
Model Interpretation ◽  
Mechanics Model ◽  
Thompson’S Group

In a recent paper, Jones introduced a correspondence between elements of the Thompson group [Formula: see text] and certain graphs/links. It follows from his work that several polynomial invariants of links, such as the Kauffman bracket, can be reinterpreted as coefficients of certain unitary representations of [Formula: see text]. We give a somewhat different and elementary proof of this fact for the Kauffman bracket evaluated at certain roots of unity by means of a statistical mechanics model interpretation. Moreover, by similar methods we show that, for some particular specializations of the variables, other familiar link invariants and graph polynomials, namely the number of [Formula: see text]-colorings and the Tutte polynomial, can be viewed as positive definite functions on [Formula: see text].

Polynomial invariants via odd parities for virtual link diagrams

2017 ◽  
Vol 26 (04) ◽  
pp. 1750021 ◽  
Author(s):  
Young Ho Im ◽  
Sera Kim ◽  
Kyoung Il Park
Keyword(s):  
Virtual Link ◽  
Polynomial Invariants ◽  
Odd Index ◽  
Link Diagrams ◽  
Virtual Links ◽  
Original Index

We introduce the odd index polynomial and the odd arrow polynomial for virtual links which are different from the original index polynomial and arrow polynomial.

Information geometry of Busemann-barycenter for probability measures

2015 ◽  
Vol 26 (06) ◽  
pp. 1541007 ◽  
Author(s):  
Mitsuhiro Itoh ◽  
Hiroyasu Satoh
Keyword(s):  
Natural Extension ◽  
Information Geometry ◽  
Acta Math ◽  
Space Structure ◽  
Hadamard Manifold ◽  
Probability Measures ◽  
Positive Density ◽  
Ideal Boundary ◽  
Push Forward ◽  
The Ideal

Using Busemann function of an Hadamard manifold X we define the barycenter map from the space 𝒫+(∂X, dθ) of probability measures having positive density on the ideal boundary ∂X to X. The space 𝒫+(∂X, dθ) admits geometrically a fiber space structure over X from Fisher information geometry. Following the arguments in [E. Douady and C. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math.157 (1986) 23–48; G. Besson, G. Courtois and S. Gallot, Entropies et rigidités des espaces localement symétriques de coubure strictement négative, Geom. Funct. Anal.5 (1995) 731–799; Minimal entropy and Mostow's rigidity theorems, Ergodic Theory Dynam. Systems16 (1996) 623–649], we exhibit that under certain geometrical hypotheses a homeomorphism Φ of the ideal boundary ∂X induces, by the aid of push-forward, an isometry of X whose extension is Φ.

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