Fields 2009 Finite Type Invariants Proposal: Difference between revisions
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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. |
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==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.
What is a Universal Finite Type Invariant?
Given a graph ("the skeleton"), let denote the set of all "knottings" of - the set of all embeddings of into considered modulo isotopy. So if is a circle, is an ordinary knot. If it is a union of circles, is a link, and if it is, say, a tetrahedron, 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 , one for each skeleton , valued in some specific family of linear spaces one for each . The spaces 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.