Difference between revisions of "Fields 2009 Finite Type Invariants Proposal"

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==What is a Universal Finite Type Invariant?==
 
==What is a Universal Finite Type Invariant?==
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[[Image:A Knotted Tetrahedron.png|thumb|right|120px]]
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Given a graph <math>\Gamma</math> ("the skeleton"), let <math>{\mathcal K}(\Gamma)</math> denote the set of all "knottings" of <math>\Gamma</math> - the set of all embeddings of <math>\Gamma</math> into <math>{\mathbb R}^3</math> considered modulo isotopy. So if <math>\Gamma</math> is a circle, <math>{\mathcal K}(\Gamma)</math> is an ordinary knot. If it is a union of circles, <math>{\mathcal K}(\Gamma)</math> is a link, and if it is, say, a tetrahedron, <math>{\mathcal K}(\Gamma)</math> will contain, for example, the knotted graph shown on the right.
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A universal finite type invariant (using a rather broad definition) is a family of invariants <math>\{Z_\Gamma:{\mathcal K}(\Gamma)\to{\mathcal A}(\Gamma)\}</math>, one for each skeleton <math>\Gamma</math>, valued in some specific family of linear spaces <math>{\mathcal A}(\Gamma)</math> one for each <math>\Gamma</math>. The spaces <math>{\mathcal A}(\Gamma)</math> 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==
 
==Some History==

Latest revision as of 14:12, 14 August 2006

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

Contents

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