26 research outputs found
Bose-Mesner Algebras attached to Invertible Jones Pairs
In 1989, Vaughan Jones introduced spin models and showed that they could be
used to form link invariants in two different ways--by constructing
representations of the braid group, or by constructing partition functions.
These spin models were subsequently generalized to so-called 4-weight spin
models by Bannai and Bannai; these could be used to construct partition
functions, but did not lead to braid group representations in any obvious way.
Jaeger showed that spin models were intimately related to certain association
schemes. Yamada gave a construction of a symmetric spin model on vertices
from each 4-weight spin model on vertices.
In this paper we build on recent work with Munemasa to give a different proof
to Yamada's result, and we analyse the structure of the association scheme
attached to this spin model.Comment: 23 page
Four-Weight Spin Models and Jones Pairs
We introduce and discuss Jones pairs. These provide a generalization and a
new approach to the four-weight spin models of Bannai and Bannai. We show that
each four-weight spin model determines a ``dual'' pair of association schemes
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
Spin Models, Association Schemes and the Nakanishi–Montesinos Conjecture
AbstractA 3-transformation of a link is a local change which replaces two strings that are three times half twisted around each other by two untwisted strings (and vice versa). The Nakanishi–Montesinos (NM) conjecture asserts that this 3-transformation can unknot any link. We introduce the notion of the NM-spin model, which gives a link invariant preserved by 3-transformation. We try to classify such spin models and determine the corresponding link invariant. It is proved that the dimension of the Bose–Mesner algebra generated by the spin model is ≤4. For dimension 1 and 2, there is no such spin model, but for dimension 3, there exists a unique one. Its link invariant is a non-trivial specialization of the Kauffman polynomial, but does not distinguish trivial links from the others, and hence cannot disprove the NM conjecture. For dimension 4, we give a family of NM-spin models. The corresponding link invariant is identified and does not distinguish trivial links from the others. Strong regularity and triple regularity of the Bose–Mesner algebra and its fusions are studied
Studies on non-amorphous association schemes and spin models
Tohoku University宗政昭弘課
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page