arXiv research

Same-genus Seifert surfaces for non-split alternating links are smoothly isotopic.

Diagrammatic method characterizes non-split surfaces in 3-sphere.

We study the Kakimizu complex of a split link. As part of this, we also study Seifert surfaces and the Kakimizu complex for a non-split link in a 3-ball. In addition, we show that a simplex of the Kakimizu complex of a non-split link can be realised in an essentially unique way.

The paper solves conditions for metacyclic actions on surfaces, including upper bounds and subgroup classifications.

We give a complete proof of results announced by Hirasawa and Sakuma describing explicitly the Kakimizu complex of a non-split, prime, special, alternating link.

We demonstrate an obstruction to finding certain splittings of four-manifolds along sufficiently twisted circle bundles over Riemann surfaces, arising from Seiberg-Witten theory. These obstructions are used to show a non-splitting result for algebraic surfaces of general type.

We show that any closed incompressible surface in the complement of a positive knot is algebraically non-split from the knot, positive knots cannot bound non-free incompressible Seifert surfaces and that the splitability and the primeness of positive knots and links can be seen from their positive diagrams.

Non-split almost complex supermanifolds and non-split Riemannian supermanifolds are studied. The first obstacle for a splitting is parametrized by group orbits on an infinite dimensional vector space. Further it is shown that non-split structures appear in the first case as deformations of a split reduction and in the …

Let $L$ be a non-split prime alternating link with $n>0$ crossings. We show that for each fixed $g$, the number of genus-$g$ Seifert surfaces for $L$ is bounded by an explicitly given polynomial in $n$. The result also holds for all spanning surfaces of fixed Euler characteristic. Previously known bounds were exponenti…

Study non-split supermanifolds from complex manifolds.

Meridian lemma extended to fully alternating links in thickened surfaces.

Classifies involutions on alternating prime non-split links.

We show that $|MS(L_1 # L_2)|=|MS(L_1)|\times|MS(L_2)|\times\mathbb{R}$ when $L_1$ and $L_2$ are any non-split and non-fibred links. Here $MS(L)$ denotes the Kakimizu complex of a link $L$, which records the taut Seifert surfaces for $L$. We also show that the analogous result holds if we study incompressible Seifert s…

This paper shows how to create surface-links with many triple points.

Characterizes alternating links in thickened surfaces using Gordon-Litherland pairing.

We prove that every embedding of $K_{2n+1,2n+1}$ into $\R^3$ contains a non-split link of $n$-components. Further, given an embedding of $K_{2n+1,2n+1}$ in $\R^3$, every edge of $K_{2n+1,2n+1}$ is contained in a non-split $n$-component link in $K_{2n+1,2n+1}$.

This paper computes Kakimizu complexes for all 11 crossing prime alternating knots.

In this paper, we study gravitational instantons (i.e., complete hyperkäler 4-manifolds with faster than quadratic curvature decay). We prove three main theorems: 1.Any gravitational instanton must have known end----ALE, ALF, ALG or ALH. 2.In ALG and ALH-non-splitting cases, it must be biholomorphic to a compact comple…

Study shows hyperbolic links are not common among prime links.

Construct non-split 2-component links in $S^4$ to produce exotic pairs of 4-manifolds

Menasco showed that a non-split, prime, alternating link that is not a 2-braid is hyperbolic in $S^3$. We prove a similar result for links in closed thickened surfaces $S \times I$. We define a link to be fully alternating if it has an alternating projection from $S\times I$ to $S$ where the interior of every complemen…

The paper explores SKT, balanced, and generalized Kähler structures on specific Lie groups.

Standard position for surfaces extended to weakly generalized alternating links.

It is a classical result that any complex analytic Lie supergroup $\mathcal{G}$ is split \cite{kosz}, that is its structure sheaf is isomorphic to the structure sheaf of a certain vector bundle. However, there do exist non-split complex analytic homogeneous supermanifolds. We study the question how to find out whether …

We introduce a version of Khovanov homology for alternating links with marking data, $ω$, inspired by instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology introduced in \cite{KM_unknot} for this marked Khovanov homology collapses on the $E_2$ page fo…

We use the theory of oriented matroids to show that any linear embedding of $K_9$, the complete graph on nine vertices, contains a non-split link with three components.

We study intrinsically linked graphs where we require that every embedding of the graph contains not just a non-split link, but a link that satisfies some additional property. Examples of properties we address in this paper are: a two component link with lk(A,L) = k2^r, k not 0, a non-split n-component link where all l…

It is a well known result from Thistlethwaite that the Jones polynomial of a non-split alternating link is alternating. We find the right generalization of this result to the case of non-split alternating tangles. More specifically: the Jones polynomial of tangles is valued in a certain skein module, we describe an alt…

We construct an infinite family of topologically slice 2--component boundary links $\ell_i$, none of which is smoothly concordant to a split link, such that $g_4(\ell_i)=i$.

The study shows involutory quandles of certain links are not left-orderable.

In this paper we investigate the unlinking numbers of 10-crossing links. We make use of various link invariants and explore their behaviour when crossings are changed. The methods we describe have been used previously to compute unlinking numbers of links with crossing number at most 9. Ultimately, we find the unlinkin…

In this paper, we explore the fixed point theory of $n$-valued maps using configuration spaces and braid groups, focussing on two fundamental problems, the Wecken property, and the computation of the Nielsen number. We show that the projective plane (resp.\ the $2$-sphere ${\mathbb S}^{2}$) has the Wecken property for …

Study shows how Khovanov homology behaves for split links and cobordisms.

Augmented alternating links are links obtained by adding trivial components that bound twice-punctured disks to non-split reduced non-2-braid prime alternating projections. These links are known to be hyperbolic. Here, we extend to show that generalized augmented alternating links, which allow for new trivial component…

New non-Kähler examples of generalized Kähler manifolds constructed via mapping tori.

The paper develops a theory of $C^\infty$-superrings and their superschemes.

We prove that every $\mathbb{Z}_2$H-thin link has no $2^k$-torsion for $k>1$ in its Khovanov homology. Together with previous results by Eun Soo Lee and the author, this implies that integer Khovanov homology of non-split alternating links is completely determined by the Jones polynomial and signature. Our proof is bas…

The paper proves limitations on actions of a specific group on spheres.

In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index $b(L)$ and arc index $α(L)$ for any knot or non-split link $L$, which are $b(L) \leq \frac{1}{2} c(L) +…

We partially solve the conjecture by A.Shumakovitch about torsion in the Khovanov homology of prime, non-split links in S^3. We give a size restriction on the Khovanov homology of almost alternating links. We relate the Khovanov homology of the connected sum of a link diagram and the Hopf link with the Khovanov homolog…

New foliations found in 3D spaces from positive braids.

We describe a "concentration on the diagonal" condition on the Khovanov complex of tangles, show that this condition is satisfied by the Khovanov complex of the single crossing tangles, and prove that it is preserved by alternating planar algebra compositions. Hence, this condition is satisfied by the Khovanov complex …

Flapan--Naimi--Pommersheim showed that every spatial embedding of $K_{10}$, the complete graph on ten vertices, contains a non-split three-component link; that is, $K_{10}$ is intrinsically triple-linked in $\mathbb{R}^3$. The work of Bowlin--Foisy and Flapan--Foisy--Naimi--Pommersheim extended the list of known intrin…

Let $D$ be a reduced alternating diagram of a non-split link $L$ and $\tilde{L}$ be the link whose diagram is obtained from $D$ by a crossing change. If $\tilde{L}$ is alternating, then $c(\tilde{L})\leq c(L)-2$. In this paper we explore when $c(\tilde{L})=c(L)-2$ holds and obtain a simple sufficient and necessary cond…

We propose a model in which a spliced vector bundle (with an arbitrary number of gauge structures in the splice) possesses a geometry which do not split. The model employs connection 1-forms with values in a space-product of Lie algebras, and therefore interlaces the various gauge structures in a non-trivial manner. Sp…

Classifies small links in an unmarked solid torus.

Study on positive scalar curvature and its impact on Ricci limit spaces.

The study improves inequalities for link diagrams and introduces weak rectangular diagrams.