Braid group, knot theory and statistical mechanics

Braid Ordering and Knot Genus

The genus of knots is a one of the fundamental invariant and can be seen as a complexity of knots. In this note, we give a lower bound of genus using Dehornoy floor, which is a measure of complexity of braids in terms of braid ordering.

Statistical Mechanics and Information Theory

Definition 1 (Spin Excess) For a system with N particles with spins ±m, the spin excess is given by how many particles more than N/2 have spin +m, i.e. S = N↑ − N2 where N↑ is the number of particles with spin +m. From Definition 1 and Equation (1) it follows that the total magnetic moment is given by M = 2mS. Theoretically, one could calculate the system’s magnetic moment at some point in time...

Quantum Mechanics and Group Theory

where xi is the the Cartesian coordinate of a point particle of mass m, Fi is the external force applied to the particle, and the force is assumed to be conservative, the negative gradient of a potential energy function V (x). Newtonian mechanics for rigid bodies can be formulated in a similar way, and this approach can even be extended to relativistic point mechanics. The calculus of variation...

Knot and Braid Invariants from Contact Homology

We introduce topological invariants of knots and braid conjugacy classes, in the form of differential graded algebras, and present an explicit combinatorial formulation for these invariants. The algebras conjecturally give the relative contact homology of certain Legendrian tori in fivedimensional contact manifolds. We present several computations and derive a relation between the knot invarian...

Conformal Field Theory and Statistical Mechanics∗