Related papers: Covering moves and Kirby calculus

We show that for any n > 3 there exists an equivalence functor from the category of n-fold connected simple coverings of B^3 x [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, to the category Chb^{3+1} of…

This paper is the second part of our work on 4-dimensional 2-handlebodies. In the first part (arXiv:math.GT/0407032) it is shown that up to certain set of local moves, connected simple coverings of B^4 branched over ribbon surfaces,…

We prove the existence of a finite set of moves sufficient to relate any two representations of the same 3-manifold as a 4-fold simple branched covering of S^3. We also prove a stabilization result: after adding a fifth trivial sheet two…

Covering moves relate colored link diagrams appearing as the branch sets of simple branched coverings of $S^3$ by the same 3-manifold. We provide a complete set of covering moves on plat closures of braids in each fixed degree $d \geq 4$,…

A combinatorial presentation of closed orientable 3-manifolds as bi-tricolored links is given together with two versions of a calculus via moves to manipulate bi-tricolored links without changing the represented manifold. That is, we…

Kirby diagrams for smooth four-dimensional manifolds typically depict only the 1- and 2-handles, omitting the 3-handles. In this work, we undertake a study of 3-handle attachments and provide tools to explicitly include them in handle…

We recall an extension of Kirby's Calculus on non-simply connected 3-manifolds given in [FR], and the surgery calculus of bridged links from [Ke], which involves only local moves. We give a short combinatorial proof that the two calculi are…

Round handles are affiliated with smooth 4-manifolds in two major ways: 5-dimensional round handles appear extensively as the building blocks in cobordisms between 4-manifolds, whereas 4-dimensional round handles are the building blocks of…

We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe one-handle attachements. As opposed to the usual link calculi of Kirby and others this description uses only…

We consider oriented knots and links in a handlebody of genus $g$ through appropriate braid representatives in $S^3$, which are elements of the braid groups $B_{g,n}$. We prove a geometric version of the Markov theorem for braid equivalence…

We study simple wrinkled fibrations, a variation of the simplified purely wrinkled fibrations introduced by Williams, and their combinatorial description in terms of surface diagrams. We show that simple wrinkled fibrations induce handle…

We construct an orientable ribbon surface F in B^4, which is universal in the following sense: any compact orientable pl 4-manifold having a handle decomposition with 0-, 1- and 2-handles can be represented as a cover of B^4 branched over…

We provide new branched covering representations for bounded and/or non-compact 4-manifolds, which extend the known ones for closed 4-manifolds. Assuming $M$ to be a connected oriented PL 4-manifold, our main results are the following: (1)…

We show that there exists an algorithm that takes as input two closed, simply connected, topological 4-manifolds and decides whether or not these 4-manifolds are homeomorphic. In particular, we explain in detail how closed, simply…

Alexander's and Markov's theorems state that any link type in $R^3$ is represented by a closed braid and that such representations are related by some elementary operations called Markov moves. We generalize the notion of a braid to that in…

We provide a complete set of moves relating any two Lefschetz fibrations over the disk having as their total space the same 4-dimensional 2-handlebody up to 2-equivalence. As a consequence, we also obtain moves relating diffeomorphic…

In this paper, we give a new direct proof of a result by Bobtcheva and Piergallini that provides finite algebraic presentations of two categories, denoted $3\mathrm{Cob}$ and $4\mathrm{HB}$, whose morphisms are manifolds of dimension $3$…

We show that only finitely many links in a closed 3-manifold share the same complement, up to twists along discs and annuli. Using the same techniques, we prove that by adding 2-handles on the same link we get only finitely many smooth…

Let $B_n$ denote the classical braid group on $n$ strands and let the {\em mixed braid group} $B_{m,n}$ be the subgroup of $B_{m+n}$ comprising braids for which the first $m$ strands form the identity braid. Let…

First steps towards a classification of irreducible symplectic 4-folds whose integral 2-cohomology with 4-tuple cup product is isomorphic to that of Hilb^2(K3). We prove that any such 4-fold deforms to an irreducible symplectic 4-fold of…