Three-dimensional integrable models and associated tangle invariants

We find that the Boltzmann weight of the three-dimensional Baxter-Bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube and the Wu-Kadanoff duality between the cube and vertex type tetrahedron equations is obtained explicitly for the Baxter-Bazhanov model. Then a three-dimensional vertex model is obtained by considering the symmetry property of the weight function, which is corresponding to the threedimensional Baxter-Bazhanov model. The vertex type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube. And we write down the symmetry relations of the weight functions under the actions of the symmetry group G of the cube. The six angles with a constrained condition, appeared in the tetrahedron equation, can be regarded as the six spectrums connected with the six spaces in which the vertex type tetrahedron equation is defined. 1 Recently big progress has been made in three-dimensional integrable models in statistical mechanics. Bazhanov and Baxter [1] introduced the Interaction-Round-a-Cube (IRC) model which is the generalization of N = 2 Zamolodchikov model [2]. Kashaev et al [3] showed that the Boltzmann weights of the Baxter-Bazhanov model satisfy the cube type tetrahedron equation by introducing the star-square relation for which the connection is found [4] with the choral Potts model. The restricted star-triangle relation and the star-star relation of this model have been discussed in detail in Refs [5, 6, 7, 8]. And they connected with the quantum dilogarithm [9] and the shift operator in discrete space-time picture [10, 11]. Then the new series of the three-dimensional integrable lattice models were presented by Mangazeev et al [12] of which the weight functions satisfy modified tetrahedron equation [13]. Recently, Cerchiai et al studied the Baxter-Bazhanov model from the point of link theory and given the representations of the braid group if some suitable spectral limits are taken. In red. [14] Korepanov got the solution of vertex tetrahedron equation with the spin variables taking N = 2 values, which leads to a commuting family of transfermatrices. From the respect of the scattering process Hietarinta discussed the three corresponding tetrahedron equations in which the Frenkel-Moore equation was fitted

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Three-dimensional integrable models and associated tangle invariants

Cited by 2 publications

References 24 publications

Knot invariants associated with a particularN→∞ continuous limit of the Baxter-Bazhanov model

Knot invariants associated with a particularN→∞ continuous limit of the Baxter-Bazhanov model

Three-dimensional vertex model in statistical mechanics from Baxter-Bazhanov model

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