occasional meanderings in physics' brave new world
- Name: Kea
- Location: New Zealand
Marni D. Sheppeard
Tuesday, July 03, 2007
C. S. Peirce is probably best known today for his work on semiotics, based on his notion of triple for which the symbols were but one component. Let's just recall today his simple representation of Boolean complement, indicated by a circle marking the boundary between A and not A. In Boolean logic it follows that a double circle should act as an identity operation. This looks just like the holographic cylinder, which we know is the identity in a category of cobordisms! Sticking to disc operads gives a large class of algebras (and connections to many, many things) but it may be necessary to consider also intersecting discs. Each Jones diagram, shown on the left below, includes internal lines that may not intersect, but when we overlap these big discs we make a choice as to their vertical ordering, and a line from one disc may cross over a line from another, thereby introducing the possibility of braiding for strands from different discs. Note that the three dimensionality of braids is reduced here by viewing the time direction only in a countable number of slices. Not all knots can be ordered in this way, because a strand may weave both under and over another one, which would require an ill defined ordering of discs. However, the basic $B_3$ elements of Bilson-Thompson diagrams are examples of ordered braids.