Kirby diagrams and 5-colored graphs representing compact 4-manifolds

Abstract

AbstractIt is well-known that in dimension 4 any framed link (L, c) uniquely represents the PL 4-manifold$$M^4(L,c)$$M4(L,c)obtained from$${\mathbb {D}}^4$$D4by adding 2-handles along (L, c). Moreover, if trivial dotted components are also allowed (i.e. in case of aKirby diagram$$(L^{(*)},d)$$(L(∗),d)), the associated PL 4-manifold$$M^4(L^{(*)},d)$$M4(L(∗),d)is obtained from$${\mathbb {D}}^4$$D4by adding 1-handles along the dotted components and 2-handles along the framed components. In this paper we study the relationships between framed links and/or Kirby diagrams and the representation theory of compact PL manifolds by edge-colored graphs: in particular, we describe how to construct algorithmically a (regular) 5-colored graph representing$$M^4(L^{(*)},d)$$M4(L(∗),d), directly “drawn over” a planar diagram of$$(L^{(*)},d)$$(L(∗),d), or equivalently how to algorithmically obtain a triangulation of$$M^4(L^{(*)},d)$$M4(L(∗),d). As a consequence, the procedure yields triangulations for any closed (simply-connected) PL 4-manifold admitting handle decompositions without 3-handles. Furthermore, upper bounds for both the invariants gem-complexity and regular genus of$$M^4(L^{(*)},d)$$M4(L(∗),d)are obtained, in terms of the combinatorial properties of the Kirby diagram.