A Polynomial Invariant of Graphs On Orientable Surfaces

2001 ◽  
Vol 83 (3) ◽  
pp. 513-531 ◽  
Author(s):  
Béla Bollobás ◽  
Oliver Riordan
Keyword(s):  
Polynomial Invariant ◽  
Orientable Surfaces

A Tutte Polynomial for Maps

2018 ◽  
Vol 27 (6) ◽  
pp. 913-945 ◽  
Author(s):  
ANDREW GOODALL ◽  
THOMAS KRAJEWSKI ◽  
GUUS REGTS ◽  
LLUÍS VENA
Keyword(s):  
Finite Group ◽  
Tutte Polynomial ◽  
Chromatic Polynomial ◽  
Polynomial Invariant ◽  
Flow Polynomial ◽  
Orientable Surfaces

We follow the example of Tutte in his construction of the dichromate of a graph (i.e. the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, the Bollobás–Riordan polynomial and the Krushkal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking values in any given finite group. Other evaluations include the number of quasi-forests.

scholarly journals Invariants of Stable Maps between Closed Orientable Surfaces

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 215
Author(s):  
Catarina Mendes de Jesus S. ◽  
Pantaleón D. Romero
Keyword(s):  
Point Of View ◽  
Stable Maps ◽  
Smooth Maps ◽  
Orientable Surfaces

In this paper, we will consider the problem of constructing stable maps between two closed orientable surfaces M and N with a given branch set of curves immersed on N. We will study, from a global point of view, the behavior of its families in different isotopies classes on the space of smooth maps. The main goal is to obtain different relationships between invariants. We will provide a new proof of Quine’s Theorem.

scholarly journals FROM RIBBON CATEGORIES TO GENERALIZED YANG–BAXTER OPERATORS AND LINK INVARIANTS (AFTER KITAEV AND WANG)

2013 ◽  
Vol 24 (01) ◽  
pp. 1250126 ◽  
Author(s):  
SEUNG-MOON HONG
Keyword(s):  
The Other ◽  
Polynomial Invariant ◽  
Link Invariants ◽  
Fusion Categories ◽  
New Family ◽  
Kauffman Polynomial ◽  
Twist Structure

We consider two approaches to isotopy invariants of oriented links: one from ribbon categories and the other from generalized Yang–Baxter (gYB) operators with appropriate enhancements. The gYB-operators we consider are obtained from so-called gYBE objects following a procedure of Kitaev and Wang. We show that the enhancement of these gYB-operators is canonically related to the twist structure in ribbon categories from which the operators are produced. If a gYB-operator is obtained from a ribbon category, it is reasonable to expect that two approaches would result in the same invariant. We prove that indeed the two link invariants are the same after normalizations. As examples, we study a new family of gYB-operators which is obtained from the ribbon fusion categories SO (N)2, where N is an odd integer. These operators are given by 8 × 8 matrices with the parameter N and the link invariants are specializations of the two-variable Kauffman polynomial invariant F.

Stable bi-maps from closed orientable surfaces to R x R

2021 ◽  
Author(s):  
Catarina Mendes de Jesus ◽  
Erica Boizan Batista ◽  
João Carlos Ferreira Costa
Keyword(s):  
Orientable Surfaces

Entanglement in four qubit states: Polynomial invariant of degree 2, genuine multipartite concurrence and one-tangle

2019 ◽  
Vol 383 (8) ◽  
pp. 707-717 ◽  
Author(s):  
M.A. Jafarizadeh ◽  
M. Yahyavi ◽  
N. Karimi ◽  
A. Heshmati
Keyword(s):  
Polynomial Invariant ◽  
Qubit States
Sign in / Sign up

Export Citation Format

Share Document