scholarly journals Polynomial invariants of graphs on surfaces

10.4171/qt/35 ◽  
2013 ◽  
Vol 4 (1) ◽  
pp. 77-90 ◽  
Author(s):  
Ross Askanazi ◽  
Sergei Chmutov ◽  
Charles Estill ◽  
Jonathan Michel ◽  
Patrick Stollenwerk
Keyword(s):  
Polynomial Invariants ◽  
Graphs On Surfaces

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.

Jones polynomial invariants for knots and satellites

1991 ◽  
Vol 109 (1) ◽  
pp. 83-103 ◽  
Author(s):  
H. R. Morton ◽  
P. Strickland
Keyword(s):  
Quantum Group ◽  
Linear Combination ◽  
Simple Formula ◽  
Jones Polynomial ◽  
Polynomial Invariants ◽  
Satellite Link ◽  
Representation Ring ◽  
Parallel Links ◽  
Jones Polynomials ◽  
Special Case

AbstractResults of Kirillov and Reshetikhin on constructing invariants of framed links from the quantum group SU(2)q are adapted to give a simple formula relating the invariants for a satellite link to those of the companion and pattern links used in its construction. The special case of parallel links is treated first. It is shown as a consequence that any SU(2)q-invariant of a link L is a linear combination of Jones polynomials of parallels of L, where the combination is determined explicitly from the representation ring of SU(2). As a simple illustration Yamada's relation between the Jones polynomial of the 2-parallel of L and an evaluation of Kauffman's polynomial for sublinks of L is deduced.

scholarly journals The Jones polynomial and graphs on surfaces

2008 ◽  
Vol 98 (2) ◽  
pp. 384-399 ◽  
Author(s):  
Oliver T. Dasbach ◽  
David Futer ◽  
Efstratia Kalfagianni ◽  
Xiao-Song Lin ◽  
Neal W. Stoltzfus
Keyword(s):  
Jones Polynomial ◽  
Graphs On Surfaces

On the polynomial invariants of the elasticity tensor

1994 ◽  
Vol 34 (2) ◽  
pp. 97-110 ◽  
Author(s):  
J. P. Boehler ◽  
A. A. Kirillov ◽  
E. T. Onat
Keyword(s):  
Elasticity Tensor ◽  
Polynomial Invariants

Coloring Graphs on Surfaces

2019 ◽  
pp. 205-222
Author(s):  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Graphs On Surfaces

scholarly journals Generalized Equidistant Chebyshev Polynomials and Alexander Knot Invariants

2018 ◽  
Vol 63 (6) ◽  
pp. 488
Author(s):  
A. M. Pavlyuk
Keyword(s):  
Chebyshev Polynomials ◽  
Polynomial Invariants ◽  
Knot Invariants ◽  
Positive Integers

We introduce the generalized equidistant Chebyshev polynomials T(k,h) of kind k of hyperkind h, where k, h are positive integers. They are obtained by a generalization of standard and monic Chebyshev polynomials of the first and second kinds. This generalization is fulfilled in two directions. The horizontal generalization is made by introducing hyperkind ℎ and expanding it to infinity. The vertical generalization proposes expanding kind k to infinity with the help of the method of equidistant coefficients. Some connections of these polynomials with the Alexander knot and link polynomial invariants are investigated.

scholarly journals Face-Width of Pfaffian Braces and Polyhex Graphs on Surfaces

10.37236/3540 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Dong Ye ◽  
Heping Zhang
Keyword(s):  
Bipartite Graph ◽  
Polynomial Time ◽  
Perfect Matching ◽  
Cartesian Product ◽  
Perfect Matchings ◽  
Face Width ◽  
Graphs On Surfaces ◽  
Positive Genus ◽  
Pfaffian Graph

A graph $G$ with a perfect matching is Pfaffian if it admits an orientation $D$ such that every central cycle $C$ (i.e. $C$ is of even size and $G-V(C)$ has a perfect matching) has an odd number of edges oriented in either direction of the cycle. It is known that the number of perfect matchings of a Pfaffian graph can be computed in polynomial time. In this paper, we show that every embedding of a Pfaffian brace (i.e. 2-extendable bipartite graph)  on a surface with a positive genus has face-width at most 3.  Further, we study Pfaffian cubic braces and obtain a characterization of Pfaffian polyhex graphs: a polyhex graph is Pfaffian if and only if it is either non-bipartite or isomorphic to the cube, or the Heawood graph, or the Cartesian product $C_k\times K_2$ for even integers $k\ge 6$.

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