Kirby diagrams and 5-colored graphs representing compact 4-manifolds
2021 09 06
Knots and representation theory
It is well-known that any framed link (L,c) uniquely represents the 3-manifold M3(L,c) obtained from S3 by Dehn surgery along (L,c), as well as the PL 4-manifold M4(L,c) obtained from D4 by adding 2-handles along (L,c). Moreover, if trivial dotted components are also allowed (i.e. in case of a Kirby diagram (L(*),c)), the associated PL 4-manifold M4(L(*),c) is obtained from D4 by adding 1-handles along the dotted components and 2-handles 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 edge-colored graphs: in particular, we describe how to construct a 5-colored 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 graph-defined invariants gem-complexity and generalized regular genus of the associated 4-manifold.
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Meeting ID: 818 6674 5751