

Knots and Representation Theory
September 6, 2021 18:30, Moscow






Kirby diagrams and 5colored graphs representing compact 4manifolds
Maria Rita Casali^{} 
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Abstract:
It is wellknown that any framed link (L,c) uniquely represents the 3manifold M3(L,c) obtained from S3 by Dehn surgery along (L,c), as well as the PL 4manifold M4(L,c) obtained from D4 by adding 2handles along (L,c). Moreover, if trivial dotted components are also allowed (i.e. in case of a Kirby diagram (L(*),c)), the associated PL 4manifold M4(L(*),c) is obtained from D4 by adding 1handles along the dotted components and 2handles along the framed components.
In the present talk we present the relationship between framed links and/or Kirby diagrams and the so called crystallization theory, which represents compact PL manifolds of arbitrary dimension by regular edgecolored graphs: in particular, we describe how to construct a 5colored graph representing M4(L(*),c), directly “drawn over” a planar diagram of (L(*), c).
As a consequence, the combinatorial properties of Kirby diagrams yield upper bounds for both the graphdefined invariants gemcomplexity and generalized regular genus of the associated 4manifold.
Language: English

