arXiv research

A marked graph diagram is a link diagram possibly with marked $4$-valent vertices. S. J. Lomonaco, Jr. and K. Yoshikawa introduced a method of representing surface-links by marked graph diagrams. Specially, K. Yoshikawa gave local moves on marked graph diagrams, nowadays called Yoshikawa moves. It is now known that two…

Paper extends method of presenting surface-links to immersed surface-links.

New Khovanov homology for marked links collapses spectral sequence.

By using the cohomology theory of quandles, quandle cocycle invariants and shadow quandle cocycle invariants are defined for oriented links and surface-links via broken surface diagrams. By using symmetric quandles, symmetric quandle cocycle invariants are also defined for unoriented links and surface-links via broken …

The paper defines new representations and groups related to virtual links.

A. S. Lipson constructed two state models yielding the same classical link invariant obtained from the Kauffman polynomial $F(a,z)$. In this paper, we apply Lipson's state models to marked graph diagrams of surface-links, and observe when they induce surface-link invariants.

Yoshikawa [Yo] conjectured that a certain set of moves on marked graph diagrams generates the isotopy relation for surface links in ${\mathbb R}^4$, and this was proved by Swenton [S] and Kearton and Kurlin [KK]. In this paper, we find another proof of this fact for the case of 2-links (surface links with spherical com…

Paper defines new invariants for surface-links using graph diagrams and magmas.

Enhances knot counting using mosaic diagrams.

In earlier work we introduced the graph bracket polynomial of graphs with marked vertices, motivated by the fact that the Kauffman bracket of a link diagram D is determined by a looped, marked version of the interlacement graph associated to a directed Euler system of the universe graph of D. Here we extend the graph b…

Frobenius extensions play a central role in the link homology theories based upon the sl(n) link variants, and each of these Frobenius extensions may be recast geometrically via a category of marked cobordisms in the manner of Bar-Natan. Here we explore a large family of such marked cobordism categories that are releva…

We present a marked analogue of Carter and Saito's movie theorem. Our definition of marking was chosen to coincide with the markings that arise in link Floer homology. In order to deal with complications arising from certain isotopies, we define three equivalence relations for marked surfaces and work over an equivalen…

Paper discusses biquandle cohomology and invariants for surface-links.

A new mosaic system for immersed surface-links is introduced.

Let $D$ be an oriented classical or virtual link diagram with directed universe $\vec{U}$. Let $C$ denote a set of directed Euler circuits, one in each connected component of $U$. There is then an associated looped interlacement graph $L(D,C)$ whose construction involves very little geometric information about the way …

It is known that every surface-link can be presented by a marked graph diagram, and such a diagram presentation is unique up to moves called Yoshikawa moves. G. Kuperberg introduced a regular isotopy invariant, called the quantum A_2 invariant, for tangled trivalent graph diagrams. In this paper, a polynomial for a mar…

In earlier work the Kauffman bracket polynomial was extended to an invariant of marked graphs, i.e., looped graphs whose vertices have been partitioned into two classes (marked and not marked). The marked-graph bracket polynomial is readily modified to handle graphs with weighted vertices. We present formulas that simp…

Proves Yoshikawa eighth move's independence and finds minimal band moves.

The article establishes a polynomial for signed cyclic graphs and links it to checkerboard colorability.

We introduce colorings of oriented surface-links by biquasiles using marked graph diagrams. We use these colorings to define counting invariants and Boltzmann enhancements of the biquasile counting invariants for oriented surface-links. We provide examples to show that the invariants can distinguish both closed surface…

We describe a method of encoding various types of link diagrams, including those with classical, flat, rigid, welded, and virtual crossings. We show that this method may be used to encode link diagrams, up to equivalence, in a notation whose length is a cubic function of the number of 'riser marks'. For classical knots…

We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot…

A realization of a virtual link diagram is obtained by choosing over/under markings for each virtual crossing. Any realization can also be obtained from some representation of the virtual link. (A representation of a virtual link is a link diagram on an oriented 2-dimensional surface.) We prove that if a minimal genus …

This paper defines several algebras associated to an oriented surface $S$ with a finite set of marked points on the boundary. The first is the skein algebra $Sk_q(S)$, which is spanned by links in the surface which are allowed to have endpoints at the marked points, modulo several locally defined relations. The product…

In-degree quiver polynomials for surface-links computed.

We introduce a class of links strictly containing quasi-alternating links for which mod 2 reduced Khovanov homology is always thin. We compute the framed instanton homology for double branched covers of such links. Aligning certain dotted markings on a link with bundle data over the branched cover, we also provide many…

New invariants for surface-links using biquandle modules.

We construct a new family, indexed by the odd integers $N\geq 1$, of $(2+1)$-dimensional quantum field theories called {\it quantum hyperbolic field theories} (QHFT), and we study its main structural properties. The QHFT are defined for (marked) $(2+1)$-bordisms supported by compact oriented 3-manifolds $Y$ with a prop…

We state Asymptotic Expansion and Growth Rate conjectures for the Witten-Reshetikhin-Turaev invariants of arbitrary framed links in 3-manifolds, and we prove these conjectures for the natural links in mapping tori of finite-order automorphisms of marked surfaces. Our approach is based upon geometric quantisation of the…

The hexabasic book is the cone of the 1-dimensional skeleton of the union of two tetrahedra glued along a common face. The universal 3-dimensional polyhedron UP is the product of a segment and the hexabasic book. We show that any 2-dimensional link in 4-space is isotopic to a surface in UP. The proof is based on a repr…

Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk with marked points. We derive some formulas for $A_1$ and $A_2$ clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formula…

It was shown by Fomin, Shapiro and Thurston that some cluster algebras arise from orientable surfaces. Subsequently, Dupont and Palesi extended this construction to non-orientable surfaces. We link this framework to Lam and Pylyavskyy's Laurent phenomenon algebras, showing that both orientable and non-orientable unpunc…

Extends Kauffman's formula to 3-manifolds with markings.

Given integers b, c, g, and n, we construct a manifold M containing a c-component link L so that there is a bridge surface Sigma for (M,L) of genus g that intersects L in 2b points and has distance at least n. More generally, given two possibly disconnected surfaces S and S', each with some even number (possibly zero) …

This paper solves the structure of link concordance groups, proving they are infinitely generated.

New invariants for welded and extended welded knots found using multi-skein relations.

In a recent paper of Akhmedov, Etnyre, Mark and Smith, it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifer…

Let $Σ$ be a hyperbolic link with $m$ components in a 3-dimensional manifold $X$. In this paper, we will show that the moduli space of marked hyperbolic cone structures on the pair $(X, Σ)$ with all cone angle less than $2π/3$ is an $m$-dimensional open cube, parameterized naturally by the $m$ cone angles. As a corolla…

Complex systems' systemic risk linked to frustration in network structure.

Odd Khovanov homology gets a new algebraic action from super foams.

We are now witnessing the increasing availability of event stream data, i.e., a sequence of events with each event typically being denoted by the time it occurs and its mark information (e.g., event type). A fundamental problem is to model and predict such kind of marked temporal dynamics, i.e., when the next event wil…

We give a topological characterisation of alternating knot exteriors based on the presence of special spanning surfaces. This shows that alternating is a topological property of the knot exterior and not just a property of diagrams, answering an old question of Fox. We also give a characterisation of alternating link e…

The Thistlethwaite theorem is extended to knotoids and linkoids.

New model captures time and mark inter-dependence in TPPs.

Classifies markings on translation surfaces from covering constructions.

Methodology for estimating marked Hawkes processes with neural networks.

Survey on decorated marked surfaces for Calabi-Yau categories.

Study of skein modules of marked 3-manifolds and Chebyshev-Frobenius homomorphism.