Configuration Spaces: Geometry, Combinatorics and Topology

seminar: WEEKLY SEMINAR: A universal algebra for (spaces of) knots

speaker: Ryan Budney (University of Victoria)

abstract: In 1949 Schubert gave the connect-sum decomposition for knots. One way to say this is that oriented smooth embeddings of S1 in S3 taken up to ambient isotopy form a free commutative monoid under the connect-sum operation. Alternatively, let K 3,1 denote the space of smooth embeddings of ℝ in ℝ3 which agree with a fixed linear embedding outside of a ball. Then π0 K3,1 is a free commutative monoid with respect to connect-sum. There is a homotopy-associative "space-level" connect-sum mapping K3,1 × K3,1K3,1. This mapping can be enhanced to an action of the operad of 2-cubes on K3,1 and in 2006 I showed that K3,1 is a free object over the 2-cubes operad, with free generating subspace the space of prime knots PK3,1, i.e. an operadic space-level analogue of Schubert’s theorem. In 1953 Schubert generalized the connect-sum operation, creating what are known as ’satellite knots’, but unlike the connect-sum operation Schubert noticed satellite knots do not decompose in a unique way. In 1979 Larry Siebenmann noticed that Schubert’s satellite constructions fit with the JSJ-decomposition of 3-manifolds, giving the appropriate uniquess statement for Schubert’s satellite operation thought of as a decomposition of knots. In this talk I will describe a new operad, ’the splicing operad’ which encodes splicing for knots at the ’spaces of knots’ level. The main theorem is that K3,1 is free with respect to the splicing operad’s action, and the free generating subspace is the subspace of K3,1 consisting of torus and hyperbolic knots. The splicing operad itself also has a pleasant structure as an operad, although it’s not quite free.

The here.

timetable:
Wed 12 May, 16:00 - 17:00, Sala Conferenze Centro De Giorgi
documents:

Budney's slides

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