Pure Maths Colloquium: Doubly periodic textile patterns
1 December 2008 16:15 in CM 107
Grishanov, Meshkov and Omelchenko have introduced the idea of representing a fabric with a repeating (doubly periodic) pattern by a knot diagram on a torus, having made a choice of a unit cell for the repeat of the pattern. Algebraic invariants of this diagram based on the Jones polynomial were used to associate a polynomial to the fabric which was independent of the choice of unit cell, so long as a minimal choice of repeating cell was made.
Grishanov and I have recently made use of the multivariable Alexander polynomial to strengthen the information available about topological properties of the fabric. We use the term fabric to mean a doubly periodic oriented plane knot diagram, consisting of coloured strands with at worst simple double point crossings, up to the classic Reidemeister moves. A fabric gives rise to a link diagram on the torus S^1 x S^1 by choosing a repeating cell in the pattern and splicing together the strands where they cross corresponding edges to form the diagram on the torus. A link in S^3 with two further auxiliary components X and Y is constructed by placing the torus in S^3 as a standard torus and including the core curves on each side of the torus in addition to the curves forming the diagram on the torus. The multivariable Alexander polynomial of this link has many nice features which are independent of the choice of unit cell, and relate more closely to the original fabric.