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UID:4842@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20230629T140000
DTEND;TZID=Europe/Paris:20230629T150000
DTSTAMP:20240524T072506Z
URL:https://www.i2m.univ-amu.fr/evenements/tba-16-2/
SUMMARY: (...): A geometric foundation for virtual knot theory
DESCRIPTION:: The theory of knots in R^3 has both a geometric model and a d
 iagrammatic model. The geometric model views a knot as point in a space K 
 of knots\, where two knots are equivalent if they lie in the same path com
 ponent of K. Alternatively\, two knot diagrams are equivalent if they are 
 related by a finite sequence of Reidemeister moves. Although the two model
 s produce the same set of equivalence classes (i.e. the knot types)\, one 
 cannot safely dispense with one and rely exclusively on the other. The geo
 metric model is needed for Chern-Simons theory and in the derivation of fi
 nite-type invariants from the cohomology of the knot space. Practical calc
 ulation of quantum invariants\, however\, requires the diagrammatics of sk
 ein theory and quantum groups. In the mid 1990s\, Kauffman introduced a ge
 neral framework for the study of knot diagrammatics. It investigates an ex
 panded set of non-planar knot diagrams\, called virtual knot diagrams\, wh
 ich are considered equivalent up to an expanded set of Reidemeister moves.
  In this talk\, we will use sheaf theory to give a complementary geometric
  model for virtual knot theory. We define a site (VK\,J_VK) so that the Gr
 othendieck topos Sh(VK\,J_VK) of sheaves on this site can be naturally int
 erpreted as the ``space of virtual knots''. A point of the space of virtua
 l knots\, that is a geometric morphism Sets → Sh(VK\,J_VK)\, is exactly 
 a virtual knot. The virtual isotopy relation and all virtual knot invarian
 ts are likewise realizable as geometric morphisms. This gives a model for 
 virtual knot theory which is geometric in the same logical sense that clas
 sical knot theory is geometric.
CATEGORIES:Séminaire,Géométrie et Topologie de Marseille
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