Luis H. Kauffman

This talk discusses logical and topological interpretations that are at the basis of the structure of continued fractions, their arborescent generalizations, arborescent links and knots, their invariant polynomials, the Temperley Lieb algebra and the interpretations of these invariants in terms of state sums and link homology. Consider the calculus of indications as generated by the mark <> and satisfying the arithmetic rules < > < > = delta <> and <<>> = = E, an identity operator , where the space before the comma is an empty word. Add to this arithmetic a new entity X so that <X>=X and XX = E. X is a “nilpotent” self-referential element in the form. We will show that this crossing arithmetic is at the bottom of a deep topological well. The iconic relationship with topology is that if <Z> denotes the ninety degree turn of the mirror image of Z (when Z is a tangle), then <X> = X when X is a crossing. Iconic logic becomes parity calculus for counting components and detecting component changes in tangles and tangle closures. The talk will illustrate how such parity calculus applies to arborescent weaving patterns and their associated invariants.