Fields 2009 Finite Type Invariants Proposal

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This is a part of a proposal for a 2009 Knot Theory Program at the Fields Institute.


What is a Universal Finite Type Invariant?

A Knotted Tetrahedron.png

Given a graph \Gamma ("the skeleton"), let {\mathcal K}(\Gamma) denote the set of all "knottings" of \Gamma - the set of all embeddings of \Gamma into {\mathbb R}^3 considered modulo isotopy. So if \Gamma is a circle, {\mathcal K}(\Gamma) is an ordinary knot. If it is a union of circles, {\mathcal K}(\Gamma) is a link, and if it is, say, a tetrahedron, {\mathcal K}(\Gamma) will contain, for example, the knotted graph shown on the right.

A universal finite type invariant (using a rather broad definition) is a family of invariants \{Z_\Gamma:{\mathcal K}(\Gamma)\to{\mathcal A}(\Gamma)\}, one for each skeleton \Gamma, valued in some specific family of linear spaces {\mathcal A}(\Gamma) one for each \Gamma. The spaces {\mathcal A}(\Gamma) are themselves defined in terms of graphs along with some linear algebra, but since we don't need the details here, we won't show them. A certain "universality" property is expected to hold, but again, we don't need it right now so we won't discuss it.

Some History

What We Expect of Finite Type Invariants

A Brief To Do List