arXiv research

Paper constructs S-equivalent genus one knots distinguishable by Jones polynomial.

Simply-connected 4-manifolds are covered by products with a 2-torus.

In this paper, Problem 4.17 on R. Kirby's problem list is solved by constructing infinitely many aspherical 4-manifolds that are homology 4-spheres

In this note we show that there are 4-manifolds not containing Gompf nucleus $N_2$; in this way we answer Problem 4.98 of Kirby's problem list in the negative.

We give a solution to a part of Problem 1.60 in Kirby's list of open problems in topology thus answering in the positive the 1987 conjecture by J.Przytycki concerning the existence of knots without matched diagrams.

C. Gordon conjectured that a connected sum of two Heegaard splittings is stabilized if and only if one of the two factors is stabilized (Problem 3.91 in Kirby's problem list). In this paper, we shall prove this conjecture.

We solve a strong version of Problem 3.6 (D) in Kirby's list, that is, we show that for any integer $n$, there exist infinitely many mutually distinct knots such that $2$-handle additions along them with framing $n$ yield the same $4$-manifold.

We give examples of closed orientable graph 3-manifolds with fundamental group which is not a subgroup of GL(4,k) for any field k. This answers a question in the Kirby problem list from 1977 which is credited to the late William Thurston.

We prove for a large class of knots that the meridional rank coincides with the bridge number. This class contains all knots whose exterior is a graph manifold. This gives a partial answer to a question of S. Cappell and J. Shaneson, see problem 1.11 on Kirby's list.

We resolve parts (A) and (B) of Problem 1.100 from Kirby's list by showing that many nontrivial links arise as cross-sections of unknotted holomorphic disks in the four-ball. The techniques can be used to produce unknotted ribbon surfaces with prescribed cross-sections, including unknotted Lagrangian disks with nontriv…

Upper bounds for surface-links in the Yoshikawa table are estimated.

New examples show unknotting numbers aren't always additive.

Yasutaka Nakanishi asked in 1981 whether a 3-move is an unknotting operation. In Kirby's problem list, this question is called `The Montesinos-Nakanishi 3-move conjecture'. We define the n-th Burnside group of a link and use the 3rd Burnside group to answer Nakanishi's question; ie, we show that some links cannot be re…

Study torsion in Kauffman bracket skein modules of 3-manifolds.

I construct "fake algebraic curves" in $Cp^2$. More precisely, for any k>2, I construct infinitely many pairwise smoothly non-isotopic (and moreover not ambient diffeomorphic) smooth surfaces $F\subset Cp^2$ homeomorphic to a non-singular algebraic curve of degree 2k, realizing the same homology class as such a curve a…

We construct examples of non-isotrivial algebraic families of smooth complex projective curves over a curve of genus 2. This solves a problem from Kirby's list of problems in low-dimensional topology. Namely, we show that 2 is the smallest possible base genus that can occur in a 4-manifold of non-zero signature which i…

Let X be a simply-connected closed oriented 4-manifold and A an embedded surface of genus g and negative self-intersection -N. We show that for fixed genus g there is an upper bound on N if the homology class of A is divisible or characteristic. In particular, for genus zero, there is a lower bound on the self-intersec…

New 4-manifolds found without 1- and 3-handles.

Smooth surfaces in simply connected 4-manifolds yield groups with non-trivial homology.

Let M_1 and M_2 be closed, orientable 3-manifolds. Let H_i denote a Heegaard surface in M_i. We prove that if H_1 # H_2 comes from stabilizing a lower genus splitting of M_1 # M_2 then either H_1 or H_2 comes from stabilizing a lower genus splitting. If H_i and G_i are non-isotopic Heegaard surfaces in M_i, and H_i is …

We prove that for any integer $n$ there exist infinitely many different knots in $S^3$ such that $n$-surgery on those knots yields the same 3-manifold. In particular, when $|n|=1$ homology spheres arise from these surgeries. This answers Problem 3.6(D) on the Kirby problem list. We construct two families of examples, t…

Finite type for 3-manifolds with boundary.

We give a topological stability result for the action of the fundamental group of a compact manifold of negative curvature on its boundary at infinity: any nearby action of this group by homeomorphisms of the sphere is semi-conjugate to the standard boundary action. Using similar techniques we prove a global rigidity r…

Review of gem theory's interactions with Kirby diagrams and trisections.

Study shows torsion in knot module for specific Montesinos knots.

The paper sets lower bounds for a Kirby-Thompson invariant of 4-manifolds.

Kirby color defined in Khovanov homology for 4D handlebodies.

From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance inv…

The paper describes handle decompositions and Kirby diagrams for plane algebraic curves.

Note on potential Kirby move type 1 for contact surgery diagrams.

This paper proves a conjecture about trisections with a specific length.

Algorithm converts Kirby diagrams to trisection diagrams for 4-manifolds.

New bounds found for complexity of spun knots.

New examples of Kirby-Ramanujam spheres found, leading to contractible 4-manifolds.

We define the notion of a Kirby element of a ribbon category C (not necessarily semisimple). Kirby elements lead to 3-manifolds invariants. We characterize (in terms of the structure maps of some categorical Hopf algebra) a set of Kirby elements of C which is sufficiently large to recover the known quantum invariants c…

Algorithm constructs Kirby diagrams for 4D open books.

High-dimensional handlebodies are shown to be products of simpler shapes.

Sharp bounds for Kirby-Thompson invariants of knotted surfaces computed.

The study describes handle decompositions and Kirby diagrams for line arrangements.

Explains Kirby's work on 4-manifold trisections.

New applications of trace embedding lemma show exotic 4-manifolds properties.

Kirby diagrams for exotic R^4's constructed from specific knot complements.

We show how to construct a Kirby diagram for a large class of finite volume hyperbolic 4-manifolds constructed by J. Ratcliffe and S. Tschantz.

A theorem of Kirby gives a necessary and sufficient condition for two framed links in S^3 to yield orientation-preserving diffeomorphic results of surgery. Kirby's theorem is an important method for constructing invariants of 3-manifolds. In this paper, we prove a variant of Kirby's theorem for null-homologous framed l…

The paper tackles deeply slice knots via immersed curves.

Study of 3-handle attachments in 4D manifolds using Kirby calculus.

The paper connects Kirby diagrams and 5-colored graphs to represent 4-manifolds.

New contact Kirby moves complete the set for contact surgery diagrams.