Related papers: An Alexander polynomial for MOY graphs
The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M, and encodes much important combinatorial information…
This paper is a self-contained development of an invariant of graphs embedded in three-dimensional Euclidean space using the Jones polynomial and skein theory. Some examples of the invariant are computed. An unlinked embedded graph is one…
We investigate the structure of the characteristic polynomial det(xI-T) of a transition matrix T that is associated to a train track representative of a pseudo-Anosov map [F] acting on a surface. As a result we obtain three new polynomial…
We consider a diagrammatic approach to investigate tame knots and links in three dimensional torus $T^3$. We obtain a finite set of generalised Reidemeister moves for equivalent links up to ambient isotopy. We give a presentation for…
We present a grapical way to describe invariants and covariants in the (4 dim) general relativity. This makes us free from the complexity of suffixes . Two new off-shell relations between (mass)$\ast\ast 6$\ invariants are obtained. These…
A braid-like isotopy for links in 3-space is an isotopy which uses only those Reidemeister moves which occur in isotopies of braids. We define a refined Jones polynomial and its corresponding Khovanov homology which are, in general, only…
The purpose of this paper is to present a certain combinatorial method of constructing invariants of isotopy classes of oriented tame links. This arises as a generalization of the known polynomial invariants of Conway and Jones. These…
Ancestral graphs can encode conditional independence relations that arise in directed acyclic graph (DAG) models with latent and selection variables. However, for any ancestral graph, there may be several other graphs to which it is Markov…
We generalized the periodic links to \emph{transitive} links in a $3$-manifold $M$. We find a complete classification theorem of transitive links in a $3$-dimensional sphere $\mathbb{R}^3$. We study these links from several different…
The $c_2$ invariant is an arithmetic graph invariant related to quantum field theory. We give a relation modulo $p$ between the $c_2$ invariant at $p$ and the $c_2$ invariant at $p^s$ by proving a relation modulo $p$ between certain…
We introduce a persistence-type invariant for finite weighted graphs based on combinatorial multivector dynamics. For each threshold parameter, a relation matrix determines a graph multivector field, whose induced directed dynamics admits a…
This article is built upon three main ideas. First, for a class of monomial ideals, it is proven that the multiplicity of an ideal equals the number of realizations of its codimension (an intuitive concept that we define later). Next, for…
Let $G=(V,E)$ be a graph and $e=uv\in E$. Define $n_u(e,G)$ be the number of vertices of $G$ closer to $u$ than to $v$. The number $n_v(e,G)$ can be defined in an analogous way. The Mostar index of $G$ is a new graph invariant defined as…
We study the structure of the stable coefficients of the Jones polynomial of an alternating link. We start by identifying the first four stable coefficients with polynomial invariants of a (reduced) Tait graph of the link projection. This…
Using the Fourier expansion of Markov traces for Ariki-Koike algebras over $\mathbb{Q}(q,u_{1},...,u_{e})$, we give a direct definition of the Alexander polynomials for mixed links. We observe that under the corresponding specialization of…
We present an accurate detailed exposition of the proof of existence of the Alexander-Conway polynomial (of links in 3-dimensional space). Other proofs were given by J. Alexander, J. Conway, V. Mantourov and L. Kauffman.
We start with a given modular invariant M of a two dimensional su(n)_k conformal field theory (CFT) and present a general method for solving the Ocneanu modular splitting equation and then determine, in a step-by-step explicit construction,…
We define some signature invariants for a class of knotted trivalent graphs using branched covers. We relate them to classical signatures of knots and links. Finally, we explain how to compute these invariants through the example of…
Given a homomorphism from a link group to a group, we introduce a $K_1$-class in another way, which is a generalization of the 1-variable Alexander polynomial. We compare the $K_1$-class with $K_1$-classes in \cite{Nos} and with…
We apply the method of orbit harmonics to the set of break divisors and orientable divisors on graphs to obtain the central and external zonotopal algebras respectively. We then relate a construction of Efimov in the context of…