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Surface invariants of finite type

De
74 pages
Surface invariants of finite type Michael Eisermann Institut Fourier, UJF Grenoble 17 September 2008 FOURIERINSTITUTfi Michael Eisermannwww-fourier.ujf-grenoble.fr/˜eiserm

  • invariant open

  • bring surfaces

  • into play

  • expansion into

  • results indicate

  • study surface invariants

  • common framework


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Surface invariants of finite type
Michael Eisermann
Institut Fourier, UJF Grenoble
17 September 2008
INSTITUT Michael Eisermannfi FOURIER www-fourier.ujf-grenoble.fr/˜eisermSummary
1 Definitions and obvious examples
Embedded and immersed surfaces
Surface invariants of finite type
The Alexander polynomial
2 The Jones polynomial of ribbon links
Skein relations
The Jones nullity
Expansion into finite type invariants
33 Finite type theory of surfaces inR
Chord diagrams on surfaces
Towards a universal invariant
Open questionsBut difficult to interpret in terms of classical topology.
Na¨ıve but natural idea: bring surfaces into play.
Study surface invariants of finite type
Analyze their interplay with links
First modest results indicate that this is successful.
Motivation
Finite-type theory of knots and links:
Common framework
Beautiful structureNa¨ıve but natural idea: bring surfaces into play.
Study surface invariants of finite type
Analyze their interplay with links
First modest results indicate that this is successful.
Motivation
Finite-type theory of knots and links:
Common framework
Beautiful structure
But difficult to interpret in terms of classical topology.First modest results indicate that this is successful.
Motivation
Finite-type theory of knots and links:
Common framework
Beautiful structure
But difficult to interpret in terms of classical topology.
Na¨ıve but natural idea: bring surfaces into play.
Study surface invariants of finite type
Analyze their interplay with linksMotivation
Finite-type theory of knots and links:
Common framework
Beautiful structure
But difficult to interpret in terms of classical topology.
Na¨ıve but natural idea: bring surfaces into play.
Study surface invariants of finite type
Analyze their interplay with links
First modest results indicate that this is successful.Summary
1 Definitions and obvious examples
Embedded and immersed surfaces
Surface invariants of finite type
The Alexander polynomial
2 The Jones polynomial of ribbon links
Skein relations
The Jones nullity
Expansion into finite type invariants
33 Finite type theory of surfaces inR
Chord diagrams on surfaces
Towards a universal invariant
Open questionsImmersed surfaces having only ribbon singularities:
(d) ribbon singularity (e) 3 ]3 (f) 61 11
3Embedded and immersed surfaces inR surfaces bounding knots or links:
(a) trivial knot, (b) trefoil knot, 3 (c) figure eight, 41 1
SeifertView, Jarke van Wijk, TUE3Embedded and immersed surfaces inR surfaces bounding knots or links:
(a) trivial knot, (b) trefoil knot, 3 (c) figure eight, 41 1
Immersed surfaces having only ribbon singularities:
(d) ribbon singularity (e) 3 ]3 (f) 61 11
SeifertView, Jarke van Wijk, TUEProposition (Fox 1962)
3 3A linkLR bounds an immersed ribbon surface #R iff it
4bounds a smoothly embedded surface ,!R without local minima.+
4Relationship with surfaces inR+
4abstract surface surface embedded inR
+
maximum
maximum
isotopy
saddle point
3h = 0 R 0