ALMOST UNKNOTTED EMBEDDINGS OF GRAPHS IN Z3 AND HIGHER DIMENSIONAL ANALOGUES

We consider the number of almost unknotted embeddings of graphs in Z3. We show that the number of such embeddings is the same, to exponential order, as the number of unknotted embeddings. We also consider some higher dimensional analogues, ie almost unknotted embeddings of surfaces which are p-dimensional analogues of Θ-graphs in Zp+2. We describe a lattice version of the spinning construction which establishes the embeddability of such surfaces in Zp+2 and show that the number of embeddings is the same, to exponential order, as the number of unknotted embeddings. The proofs of our upper bounds feature a novel application of the classical Loomis–Whitney inequality.

LATTICE FOKKER–PLANCK EQUATION

A lattice version of the Fokker–Planck equation is introduced. The resulting numerical method is illustrated through the calculation of the electric conductivity of a one-dimensional charged fluid at zero and finite-temperature.

Groups in which every subgroup is modular-by-finite

A group G is called a BCF-group if there is a positive integer κ such that |X : XG| ≤ κ for each subgroup X of G. The structure of BCF-groups has been studied by Buckley, Lennox, Neumann, Smith and Wiegold; they proved in particular that locally finite groups with the property BCF are Abelian-by-finite. As a group lattice version of this concept, we say that a group G is a BMF-group if there is a positive integer κ such that every subgroup X of G contains a modular subgroup Y of G for which the index |X : Y| is finite and the number of its prime divisors with multiplicity is bounded by κ (it is known that that such number can be characterised by purely lattice-theoretic considerations, and so it is invariant under lattice isomorphisms of groups). It is proved here that any locally finite BMF-group contains a subgroup of finite index with modular subgroup lattice.

Q-DEFORMATION OF VIRASORO ALGEBRA AND LATTICE CONFORMAL THEORIES

A natural definition of q-deformation of Virasoro and superconformal algebras is proposed. New Lie algebraic symmetries are shown to describe the lattice version of the original theory. On the classical (Poisson brackets) level these two-loop algebras are shown to be isomorphic to the Faddeev-Takhtadjan-Volkov lattice Virasoro algebra.