Tetsuo Deguchi

Exactly solvable models and knot theory

Abstract Presented is a review on theory of exactly solvable models in statistical mechanics and its application to knot theory. The Yang-Baxter relation, a sufficient condition for the solvability of models, is introduced for scattering matrices in (1 + 1)-dimensional field theory and for Boltzmann weights of vertex models and IRF models in two-dimensional statistical mechanics. A systematic study of solutions of the Yang-Baxter relation shows that there exists at least an infinite number of two-dimensional exactly solvable models in classical statistical mechanics. The result implies that each universality class has at least one exactly solvable model. A novel connection between physics and mathematics is exposed. Namely, a general theory to derive link polynomials, topological invariants for knots and links, from the exactly solvable models is presented. It is emphasized that the Yang-Baxter relation is a key to relate various new developments in recent theoretical physics.

The sl2 Loop Algebra Symmetry of the Six-Vertex Model at Roots of Unity

We demonstrate that the six vertex model (XXZ spin chain) with Δ=(q+q-1)/2 and q2N=1 has an invariance under the loop algebra of sl2 which produces a special set of degenerate eigenvalues. For Δ=0 we compute the multiplicity of the degeneracies using Jordan–Wigner techniques.

Dimension of ring polymers in bulk studied by Monte-Carlo simulation and self-consistent theory.

We studied equilibrium conformations of ring polymers in melt over the wide range of segment number N of up to 4096 with Monte-Carlo simulation and obtained N dependence of radius of gyration of chains R(g). The simulation model used is bond fluctuation model (BFM), where polymer segments bear excluded volume; however, the excluded volume effect vanishes at N-->infinity, and linear polymer can be regarded as an ideal chain. Simulation for ring polymers in melt was performed, and the nu value in the relationship R(g) proportional to N(nu) is decreased gradually with increasing N, and finally it reaches the limiting value, 1/3, in the range of N>or=1536, i.e., R(g) proportional to N(1/3). We confirmed that the simulation result is consistent with that of the self-consistent theory including the topological effect and the osmotic pressure of ring polymers. Moreover, the averaged chain conformation of ring polymers in equilibrium state was given in the BFM. In small N region, the segment density of each molecule near the center of mass of the molecule is decreased with increasing N. In large N region the decrease is suppressed, and the density is found to be kept constant without showing N dependence. This means that ring polymer molecules do not segregate from the other molecules even if ring polymers in melt have the relationship nu=1/3. Considerably smaller dimensions of ring polymers at high molecular weight are due to their inherent nature of having no chain ends, and hence they have less-entangled conformations.

Exactly Solvable Models and New Link Polynomials. II. Link Polynomials for Closed 3-Braids

Using a generalized Alexander-Conway relation derived from a three-state exactly solvable model in statistical mechanics, new invariant polynomials for knots and links are explicitly evaluated. It is shown that the invariant polynomials for closed 3-braids are obtained recursively. It is also shown that the invariant polynomials are more powerful than the Jones polynomials.

A Statistical Study of Random Knotting Using the Vassiliev Invariants

Employing the Vassiliev invariants as tools for determining knot types of polygons in 3 dimensions, we evaluate numerically the knotting probability PK(N) of the Gaussian random polygon being equivalent to a knot type K. For prime knots and composite knots we plot the knotting probability PK(N) against the number N of polygonal nodes. Taking the analogy with the asymptotic scaling behaviors of self-avoiding walks, we propose a formula of fitting curves to the numerical data. The curves fit well the graphs of the knotting probability PK(N) versus N. This agreement suggests to us that the scaling formula for the knotting probability might also work for the random polygons other than the Gaussian random polygon.

Exactly Solvable Models and New Link Polynomials. IV. IRF Models

We present a general method to construct link polynomials, invariants for knots and links, from the exactly solvable IRF (Interaction Round a Face) models in statistical mechanics which satisfy the Yang-Baxter relation.

Exactly solvable models and new link polynomials. V: Yang-Baxter operator and braid-monoid algebra

Yang-Baxter operator is shown to be a fundamental object which relates theory of solvable models to theory of knots and links. First, general properties of Yang-Baxter operators are investigated. Second, a method to construct composite Yang-Baxter operators is explicitly shown. Lastly, from Yang-Baxter operators with crossing symmetry, braid-monoid algebras are derived. It is emphasized that the factorized S -matrices and their graphical illustrations link two approaches, algebraic and combinatorial, in the knot theory.

Exactly solvable models and new link polynomials. III: Two-variable topological invariants

New link polynomials, reported in I and II of the series, are extended into those with two variables. A concept of composite string is introduced. It is shown that the composite string representation and the generalized Ocneanus trace lead to a sequence of two-variable link polynomials. In addition, algebraic aspects of the composite string representation are studied in some detail.

Quantum superalgebra Uqosp(2,2)

Abstract We construct the quantum universal enveloping algebra U q osp(2,2) and obtain the universal R -matrix for the algebra. The matrix elements of the R -matrix are explicitly calculated for the fundamental representation. This gives a new solvable model (36-vertex model) associated with the algebra.

Graded solutions of the Yang-Baxter relation and link polynomials

From a family of graded solvable models the authors derive representations of the braid group associated with the Lie superalgebra gl(M mod N) and give explicitly a general form of the Markov traces on the representations. The braid operators thus obtained satisfy the Hecke algebra. The authors construct composite solvable models and obtain link polynomials from the braid operators for the composite models.