Related papers: A bracket polynomial for graphs. III. Vertex weigh…
We show that Kauffman brackets of colored framed graphs (also known as quantum spin networks) can be renormalized to a Laurent polynomial with integer coefficients by multiplying it by a coefficient which is a product of quantum factorials…
The W-polynomial is applied in two ways to questions involving the Kauffman bracket of some families of links. First we find a geometric property of a link diagram, which is less than or equal to the twist number, that bounds the Mahler…
We introduce a new graph polynomial in two variables. This ``interlace'' polynomial can be computed in two very different ways. The first is an expansion analogous to the state space expansion of the Tutte polynomial; the significant…
In \cite{4} Kauffman and Vogel constructed a rigid vertex regular isotopy invariant for unoriented four-valent graphs embedded in three dimensional space. It assigns to each embedded graph $G$ a polynomial, denoted $[G]$, in three…
Motivated by circle graphs, and the enumeration of Euler circuits, we define a one-variable ``interlace polynomial'' for any graph. The polynomial satisfies a beautiful and unexpected reduction relation, quite different from the cut and…
We prove a connection between Schmidt-rank and weight of quadratic forms. This provides a new tool for the classification of graph states based on entanglement. Our main tool arises from a reformulation of previously known results…
The ribbon group action extends geometric duality and Petrie duality by defining two embedded graphs as twisted duals precisely when they lie within the same orbit under this group action. Twisted duality yields numerous novel properties of…
In loop quantum gravity, states of quantum geometry are represented by classes of knotted graphs, equivalent under diffeomorphisms. Thus, it is worthwhile to enumerate and distinguish these classes. This paper looks at the case of 4-regular…
We show that the adjacency matrices of the intersection graphs of chord diagrams satisfy the 2-term relations of Bar-Natan and Garoufalides [bg], and hence give rise to weight systems. Among these weight systems are those associated with…
In [3] we constructed the parity-biquandle bracket valued in {\em pictures} (linear combinations of $4$-valent graphs). We gave no example of classical links such that the parity-biquandle bracket of which is not trivial. In the present…
We extend the notion of intersection graphs for knots in the theory of finite type invariants to string links. We use our definition to develop weight systems for string links via the adjacency matrix of the intersection graph, and show…
We introduce an invariant of alternating knots and links (called here WRP), namely a pair of integer polynomials associated with their two checkerboard planar graphs from their minimal diagram. We prove that the invariant is well-defined…
S. Nelson, M. Orrison, V. Rivera {\cite{S}} modified Kauffman's construction of bracket. Their invariant $\Phi^{\beta}_X$ takes value in a finite ring $Z_2[t]/(1+t+t^3)$. In this paper, the author generalizes this invariant. The new…
In this article, we focus on the characteristic polynomial of a graph containingloops, but without multiple edges. We present a relationship between thecharacteristic polynomial of a graph with loops and the graph obtained byremoving all…
Graph polynomials encode fundamental combinatorial invariants of graphs. Their computation is investigated using tree and path decomposition frameworks, with formal definitions of treewidth, k-trees, and pathwidth establishing the…
We classify trivalent vertex-transitive graphs whose edge sets have a partition into a 2-factor composed of two cycles and a 1-factor that is invariant under the action of the automorphism group.
A matrix-weighted graph is an undirected graph with a $k\times k$ positive semidefinite matrix assigned to each edge. There are natural generalizations of the Laplacian and adjacency matrices for such graphs. These matrices can be used to…
M. Niebrzydowski and J. H. Przytycki defined a Kauffman bracket magma and constructed the invariant P of framed links in 3-space. The invariant is closely related to the Kauffman bracket polynomial. The normalized bracket polynomial is…
Given a complete graph with positive weights on its edges, we define the weight of a subset of edges as the product of weights of the edges in the subset and consider sums (partition functions) of weights over subsets of various kinds:…
For a ribbon graph $G$ we consider an alternating link $L_G$ in the 3-manifold $G\times I$ represented as the product of the oriented surface $G$ and the unit interval $I$. We show that the Kauffman bracket $[L_G]$ is an evaluation of the…