by S K Nechaev (Landau Institute for Theoretical PhysicsInstitut de Physique Nucléaire, RussiaFrance)

In this book, the author announces the class of problems called “entropy of knots” and gives an overview of modern physical applications of existing topological invariants.

He constructs statistical models on knot diagrams and braids using the representations of Jones–Kauffman and Alexander invariants and puts forward the question of limit distribution of these invariants for randomly generated knots. The relation of powers of corresponding algebraic invariants to the Lyapunov exponents of the products of noncommutative matrices is described. Also the problem of conditional joint limit distributions for “brownian bridges” on braids is discussed. Special cases of noncommutative groups PSL(2,R), PSL(2,Z) and braid groups are considered in detail.

In this volume, the author also discusses the application of conformal methods for explicit construction of topological invariants for random walks on multiconnected manifolds. The construction of these topological invariants and the monodromy properties of correlation function of some conformal theories are also discussed.

The author also considers the physical applications of “knot entropy” problem in various physical systems, focussing on polymers.

• Knot Diagrams as Disordered Spin Systems:
  • Introduction: Statistical Problems in Topology
  • Review of Abelian Problems in Statistics of Entangled Random Walks and Incompleteness of Gauss Invariant
  • Nonabelian Algebraic Knot Invariants
  • Lattice Knot Diagrams as Disordered Potts Model
  • Annealed and Quenched Realizations of Topological Disorder
• Random Walks on Local Noncommutative Groups:
  • Introduction
  • Brownian Bridges on Simplest Noncommutative Groups and Knot Statistics
  • Random Walks on Locally Free Groups
  • Brownian Bridges on Lobachevskii Plane and Products of Noncommutative Random Matrices
• Conformal Methods in Statistics of Entangled Random Walks:
  • Introduction: Random Walk with Topological Constraints
  • Construction of Nonabelian Connections for Γ₂ and PSL(2,Z) from Conformal Methods
  • Random Walk on Double Punctured Plane and Conformal Field Theory
  • Statistics of Random Walks with Topological Constraints in 2D Lattice of Obstacles
• Physical Applications:
  • Introduction: Polymer Language in Statistics of Entangled Chain-Like Objects
  • Polymer Chain in 3D Array of Obstacles: Critical Exponents for Gyration Radius
  • High Elasticity of Polymer Networks
  • Collapsed Phase of Unknotted Polymer
  • Ordering Phase Transition in Entangled “Directed Polymers”