One-Cocycles and Knot Invariants is about classical knots, i.e., smooth oriented knots in 3-space. It introduces discrete combinatorial analysis in knot theory in order to solve a global tetrahedron equation. This new technique is then used to construct combinatorial 1-cocycles in a certain moduli space of knot diagrams. The construction of the moduli space makes use of the meridian and the longitude of the knot. The combinatorial 1-cocycles are therefore lifts of the well-known Conway polynomial of knots, and they can be calculated in polynomial time. The 1-cocycles can distinguish loops consisting of knot diagrams in the moduli space up to homology. They give knot invariants when they are evaluated on canonical loops in the connected components of the moduli space. They are a first candidate for numerical knot invariants which can perhaps distinguish the orientation of knots.
Contents:
- Introduction
- The 1-Cocycles from the Conway Polynomial
- Proofs
- The Examples
- Two Forgotten Linear 1-Cocycles
- An Eclectic 1-Cocycle
Readership: This title will be particularly useful for academic researchers and PhD students.
Key Features:
- The main problem in knot theory in 3-space, namely giving a system of calculable numerical knot invariants, which distinguish all smooth oriented classical knots, is still open since the first mathematical paper about knots in 1771. This title makes an essential step towards a potential solution to this problem. So far, the construction of knot invariants in 3-space was mainly based on representation theory and hence on algebra. The book introduces combinatorial analysis as a new tool in order to study classical knots, by studying families of knot diagrams instead of individual diagrams. The basis of this new technique was laid in the previous title Polynomial One-cocycles for Knots and Closed Braids. This new book develops this technique further to its full power