A theorem of Kirby states that two framed links in the 3-sphere produce orientation-preserving homeomorphic results of surgery if they are related by a sequence of stabilization and handle-slide moves. The purpose of the present paper is twofold: First, we give a sufficient condition for a sequence of handle-slides on …
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20 results for “handle-slides”
Kirby color defined in Khovanov homology for 4D handlebodies.
problem Invariance of 4D handlebodies under Kirby moves.
method Functoriality and cabling properties of Khovanov homology, handle slide isomorphism.
result Kirby-colored Khovanov homology is invariant under handle slide moves.
We propose a means by which some categorifications can be evaluated at a root of unity. This is implemented using a suitable localization in the context of prior work by the authors on categorification of the Jones-Wenzl projectors. Within this construction we define objects, invariant under handle slides, which decate…
A knot K in the 3-sphere is said to have Property nR if, whenever K is a component of an n-component link L and some integral surgery on L produces the connected sum of n copies of S^1 x S^2, there is a sequence of handle slides on L that converts L into a 0-framed unlink. The Generalized Property R Conjecture is that …
Paper studies simplified trisections and their equivalence classes.
problem Understanding right-left equivalence of simplified -trisections.
method Analyzes simplified trisection diagrams and upper-triangular handle-slides.
result At least two simplified -trisections can be right-left equivalent without being related by automorphisms or handle-slides.
Paper disproves a theorem about Kauffman bracket skein module structure.
problem Disproving a 22-year-old theorem about Kauffman bracket skein module structure.
method Analyzing handle slidings on compressing discs in handlebodies.
result More relations found than previously predicted for connected sum of handlebodies.
Study shows links with trivial Lagrangian are slice, useful for surgery problems.
problem Understanding the slice properties of links and their implications for 4D surgery.
method Analysis of Nielsen moves, handle slides, and Seifert surfaces with trivial Lagrangian.
result A collection of links admits Seifert surfaces with trivial Lagrangian, proving their slice property.
We exhibit a finite set of local moves that connect any two surgery presentations of the same 3-manifold via framed links in the three-sphere. The moves are handle-slides and blow-downs/ups of a particular simple kind.
New contact Kirby moves complete the set for contact surgery diagrams.
problem Contact surgery diagrams and their relation to contactomorphic contact manifolds.
method Introducing lantern moves and chain moves to complete the set of contact Kirby moves.
result Two contact surgery diagrams represent contactomorphic contact manifolds if and only if they are related by a sequence of specific moves.
Khovanov homology for links in math.GT
Extends Khovanov homology to links in a specific manifold.
problem Defining Khovanov homology for links in a new manifold.
method Extends Rozansky's construction to $M^r=#^r(S^2 imes S^1)$, defining and exploring its properties.
result Recovery of WRT invariants and evaluation in skein module for null-homologous links.
Researchers compute the Kauffman bracket skein module of a specific 3-manifold.
problem Understanding the structure of Kauffman bracket skein modules for non-prime manifolds.
method Analyzing handle sliding relations to compute the module over .
result The skein module of does not split into free and torsion submodules.
Handles decompositions reveal new open book structures.
problem Understanding and manipulating open book structures on manifolds.
method Build handle decompositions and describe handle slides.
result Stabilization of open books to trivial monodromy.
The stable Andrews-Curtis conjecture in combinatorial group theory is the statement that every balanced presentation of the trivial group can be simplified to the trivial form by elementary moves corresponding to "handle-slides" together with "stabilization" moves. Schoenflies conjecture is the statement that the compl…
Kirby proved that two framed links in S^3 give orientation-preserving homeomorphic results of surgery if and only if these two links are related by a sequence of two kinds of moves called stabilizations and handle-slides. Fenn and Rourke gave a necessary and sufficient condition for two framed links in a closed, orient…
The above named paper has been withdrawn. A colleague has observed a gap in the proof of isotopy invariance, which can be repaired by reducing the coefficients (which lie in (1/6)Z) of the antisymmetric kanji with chords incident with more than one component modulo 8Z. An analogous issue arises in considering the effec…
We show that simple coverings of B^4 branched over ribbon surfaces up to certain local ribbon moves bijectively represent orientable 4-dimensional 2-handlebodies up to handle sliding and addition/deletion of cancelling handles. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over…
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, bijectively represent connected orientable 4-dimensional 2-handlebodies up to 2-deforma…
This paper outlines an approach to the non-abelian theta functions of the -Chern-Simons theory with the methods used by A. Weil for studying classical theta functions. First we translate in knot theoretic language classical theta functions, the action of the finite Heisenberg group, and the discrete Fourier tran…
Minimal moves for surfaces in 4D discovered, linking planar and spatial moves.
problem Finding the minimal set of moves for surfaces in 4D.
method Derived minimal generating set of spatial moves, translated into planar moves.
result Minimal generating set of spatial moves for surfaces in 4D.
The HKR (Hennings-Kauffman-Radford) framework is used to construct invariants of 4-thickenings of 2-dimensional CW complexes under 2-deformations (1- and 2- handle slides and creations and cancellations of 1-2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an elem…