arXiv research

We show that the Kakimizu complex of a knot may be locally infinite, answering a question of Przytycki--Schultens. We then prove that if a link $L$ only has connected Seifert surfaces and has a locally infinite Kakimizu complex then $L$ is a satellite of either a torus knot, a cable knot or a connected sum, with windin…

Scharlemann and Schultens have shown that for any pair of knots K_1 and K_2, w(K_1 # K_2) is greater than or equal to max{w(K_1),w(K_2)}. Scharlemann and Thompson have given a scheme for possible examples where equality holds. Using results of Scharlemann-Schultens, Rieck-Sedgwick and Thompson, it is shown that for K t…

Framework for reconstructing knot exterior surface complexes.

In "Width complexes for knots and 3-manifolds," Jennifer Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in the 3-sphere and the isotopies between these positions. She poses several questions about these width complexes; in particular, she asks whether the…

We use Heath's, Moriah's and Schultens's results to prove that irreducible Heegaard splittings of orientable Seifert manifolds with nonorientable base space are either vertical or horizontal.

Scharlemann and Thompson define the width of a 3-manifold M as a notion of complexity based on the topology of M. Their original definition had the property that the adjacency relation on handles gave a linear order on handles, but here we consider a more general definition due to Saito, Scharlemann and Schultens, in w…

Let k be a knot in S3. In [8], H.N. Howards and J. Schultens introduced a method to construct a manifold decomposition of double branched cover of (S3, k) from a thin position of k. In this article, we will prove that if a thin position of k induces a thin decomposition of double branched cover of (S3,k) by Howards and…

The paper provides examples of keen weakly reducible bridge spheres for links in b-bridge position.

The paper studies tunnel and bridge numbers of composite genus 2 spatial graphs.

Thin position for knots in the 3-sphere was introduced by Gabai and has been used in a variety of contexts. We conjecture an analogue to a theorem of Schubert and Schultens concerning the bridge number of satellite knots. For a satellite knot K, we use the companion torus T to provide a lower bound for w(K), proving th…

A knot k in a closed orientable 3-manifold is called nonsimple if the exterior of k possesses a properly embedded essential surface of nonnegative Euler characteristic. We show that if k is a nonsimple prime tunnel number one knot in a lens space M (where M does not contain any embedded Klein bottles), then k is a (1,1…

We develop a construction suggested by Scharlemann and Thompson to obtain an infinite family of pairs of knots $K_α$ and $K'_α$ so that $w(K_α # K'_α)=max{w(K_α), w(K'_α)}$. This is the first known example of a pair of knots such that $w(K#K')<w(K)+w(K')-2$ and it establishes that the lower bound $w(K#K')\geq max{w(K),…

We introduce a new technique for finding lower bounds on the Heegaard genus of a 3-manifold obtained by gluing a pair of 3-manifolds together along an incompressible torus or annulus. We deduce a number of inequalities, including one which implies that $t(K_1# K_2)\geq \max {t(K_1),t(K_2)}$, where $t(-)$ denotes tunnel…

A Heegaard splitting of an open 3-manifold is the partition of the manifold into two non-compact handlebodies which intersect on their common boundary. This paper proves several non-compact analogues of theorems about compact Heegaard splittings. The main theorem is: if N is a compact, connected, orientable 3-manifold …

We formulate a generalization of the volume conjecture for planar graphs. Denoting by <G, c> the Kauffman bracket of the graph G whose edges are decorated by real "colors" c, the conjecture states that, under suitable conditions, certain evaluations of <G,kc> grow exponentially as k goes to infinity and the growth rate…

In graph theory, as well as in 3-manifold topology, there exist several width-type parameters to describe how "simple" or "thin" a given graph or 3-manifold is. These parameters, such as pathwidth or treewidth for graphs, or the concept of thin position for 3-manifolds, play an important role when studying algorithmic …

New condition prevents hyperbolic spaces from matching curve complexes.

Study on complex line fields on almost-complex manifolds, proving existence conditions.

Homotopy types of curve and arc complexes are studied.

This research explores complex-valued neural networks and their implementation.

Paper introduces fat CW complexes including all closed manifolds.

The paper discusses $q$-deformations of the Aomoto complex.

Study calculates global sections on complex curves.

The paper studies lifts of complex structures on a manifold.

In this paper, we first provide an updated survey of the geometry of complex Cartan spaces. New characterizations for some particular classes of complex Cartan spaces are pointed out, e.g. Landsberg-Cartan, strongly Berwald-Cartan and others. We introduce the Cartan-Randers spaces which offer examples of Berwald-Cartan…

Study $L^2$ Hilbert complexes on complex manifolds.

The paper defines and constructs almost complex blow-ups on 4D almost complex manifolds.

Research shows arc complex is not quasi-isometric to sphere complex.

Study Hodge-de Rham numbers for almost complex 4-manifolds, extending properties from complex surfaces.

New proofs for growth series of Coxeter groups using complex structures.

In this article, we consider Cayley deformations of a compact complex surface in a Calabi--Yau four-fold. We will study complex deformations of compact complex submanifolds of Calabi--Yau manifolds with a view to explaining why complex and Cayley deformations of a compact complex surface are the same. We in fact prove …

A Sasaki-like almost contact complex Riemannian manifold is defined as an almost contact complex Riemannian manifold which complex cone is a holomorphic complex Riemannian manifold. Explicit compact and non-compact examples are given. A canonical construction producing a Sasaki-like almost contact complex Riemannian ma…

Study $Sp(n)$-orbits in complex and $Σ$-complex subspaces of Hermitian quaternionic vector spaces.

Tree complex linked to polyhedral shapes like associahedra and cyclohedra.

We show that any compact almost-complex manifold of complex dimension m can be pseudo-holomorphically embedded in R^(6m) equipped with a suitable almost-complex structure.

Geometric model for Hodge filtered complex cobordism constructed.

New calculations of topological complexity for symplectic CW-complexes.

This note constructs complex structures on specific isoparametric hypersurfaces.

Proposes a method to learn representations of higher-dimensional simplicial complexes.

We consider options that pay the complexity deficiency of a sequence of up and down ticks of a stock upon exercise. We study the price of European and American versions of this option numerically for automatic complexity, and theoretically for Kolmogorov complexity. We also consider run complexity, which is a restricte…

The paper explores complex Poisson structures on smooth functions in complex manifolds.

Proposes CXNs for neural network computations on cell complexes.

Note on connectedness of primitive disk complex.

Almost complex structures found on many homotopy complex projective spaces.

Survey on hypothetical complex structure on 6-sphere.

Estimates complex Hessian integral for complex Monge-Ampère equations.

Study cohomology of Bigolin complex on complex manifolds.

Tiny complexes share 3-5 triangles in common coverings.