10,594 research outputs found
Singular Links and Yang-Baxter State Models
We employ a solution of the Yang-Baxter equation to construct invariants for
knot-like objects. Specifically, we consider a Yang-Baxter state model for the
sl(n) polynomial of classical links and extend it to oriented singular links
and balanced oriented 4-valent knotted graphs with rigid vertices. We also
define a representation of the singular braid monoid into a matrix algebra, and
seek conditions for extending further the invariant to contain topological
knotted graphs. In addition, we show that the resulting Yang-Baxter-type
invariant for singular links yields a version of the Murakami-Ohtsuki-Yamada
state model for the sl(n) polynomial for classical links.Comment: 22 pages, many figures; this is the journal version of the pape
Unsigned state models for the Jones polynomial
It is well a known and fundamental result that the Jones polynomial can be
expressed as Potts and vertex partition functions of signed plane graphs. Here
we consider constructions of the Jones polynomial as state models of unsigned
graphs and show that the Jones polynomial of any link can be expressed as a
vertex model of an unsigned embedded graph.
In the process of deriving this result, we show that for every diagram of a
link in the 3-sphere there exists a diagram of an alternating link in a
thickened surface (and an alternating virtual link) with the same Kauffman
bracket. We also recover two recent results in the literature relating the
Jones and Bollobas-Riordan polynomials and show they arise from two different
interpretations of the same embedded graph.Comment: Minor corrections. To appear in Annals of Combinatoric
Colored Vertex Models, Colored IRF Models and Invariants of Trivalent Colored Graphs
We present formulas for the Clebsch-Gordan coefficients and the Racah
coefficients for the root of unity representations (-dimensional
representations with ) of . We discuss colored vertex
models and colored IRF (Interaction Round a Face) models from the color
representations of . We construct invariants of trivalent colored
oriented framed graphs from color representations of .Comment: 39 pages, January 199
From the Ising and Potts models to the general graph homomorphism polynomial
In this note we study some of the properties of the generating polynomial for
homomorphisms from a graph to at complete weighted graph on vertices. We
discuss how this polynomial relates to a long list of other well known graph
polynomials and the partition functions for different spin models, many of
which are specialisations of the homomorphism polynomial.
We also identify the smallest graphs which are not determined by their
homomorphism polynomials for and and compare this with the
corresponding minimal examples for the -polynomial, which generalizes the
well known Tutte-polynomal.Comment: V2. Extended versio
Jaeger's Higman-Sims state model and the B_2 spider
Jaeger [Geom. Dedicata 44 (1992), 23-52] discovered a remarkable checkerboard
state model based on the Higman-Sims graph that yields a value of the Kauffman
polynomial, which is a quantum invariant of links. We present a simple argument
that the state model has the desired properties using the combinatorial
spider [Comm. Math. Phys. 180 (1996), 109-151]
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