arXiv research

The abstract introduces a sequence of moves between two types of Heegaard diagrams for knot Floer homology.

In this paper, we characterize the sigma-adequacy of a link diagram in two ways: in terms of a certain edge subset of its Tait graph and in terms of a certain product of Tutte polynomials. Furthermore, we show that the symmetrized Tutte polynomial of the Tait graph of a link diagram can be written as a sum of these pro…

Formulae for Vassiliev invariants derived from Kauffman polynomial.

The Kauffman bracket polynomial is calculated for specific Turk's head knots.

New method finds grid diagrams for many fibered knots.

We study decomposition into simple arcs (i. e., arcs without self-intersections) for diagrams of knots and spatial graphs. In this paper, it is proved in particular that if no edge of a finite spatial graph $G$ is a knotted loop, then there exists a plane diagram $D$ of $G$ such that (i) each edge of $G$ is represented…

The paper constructs quantum invariants for knotoid diagrams.

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.

We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for…

Incorrect parity-based descriptions of realizable Gauss diagrams found, but bipartite graphs provide a valid approach.

Study minimum ribbonlength of immersed flat knots and links.

New method calculates manifold invariants using Kirby diagrams with 3-handles.

We define and study a bigraded knot invariant whose Euler characteristic is the Alexander polynomial, closely connected to knot Floer homology. The invariant is the homology of a chain complex whose generators correspond to Kauffman states for a knot diagram. The definition uses decompositions of knot diagrams: to a co…

We describe the Polyak-Viro arrow diagram formulas for the coefficients of the Conway polynomial. As a consequence, we obtain the Conway polynomial as a state sum over some subsets of the crossings of the knot diagram. It turns out to be a simplification of a special case of Jaeger's state model for the HOMFLY polynomi…

The paper proves a generalized Kauffman-Harary conjecture for prime determinant links.

The state-sum invariants for knots and knotted surfaces defined from quandle cocycles are described using the Kronecker product between cycles represented by colored knot diagrams and a cocycle of a finite quandle used to color the diagram. Such an interpretation is applied to evaluating the invariants. Algebraic inter…

The paper defines new polynomials for links and linkoids.

Fundamental weight systems identified as quantum states.

Recently, Dasbach, Futer, Kalfagianni, Lin, and Stoltzfus extended the notion of a Tait graph by associating a set of ribbon graphs (or equivalently, embedded graphs) to a link diagram. Here we focus on Seifert graphs, which are the ribbon graphs of a knot or link diagram that arise from Seifert states. We provide a ch…

MXGNet tackles visual reasoning tasks using graph neural networks.

Construct petal diagrams from simple braids to verify knot petal numbers.

Tait's flyping conjecture, stating that two reduced, alternating, prime link diagrams can be connected by a finite sequence of flypes, is extended to reduced, alternating, prime diagrams of 4-regular graphs in S^3. The proof of this version of the flyping conjecture is based on the fact that the equivalence classes wit…

Round surgery diagrams represent 3-manifolds in $S^3$.

Proves a theorem for surface links, simplifying diagrams of links.

In the author's earlier work there appeared a new way to specify any smooth closed 4-manifold by a surface diagram, which consists of an orientable surface decorated with simple closed curves. These curves are cyclically indexed, and each curve has a unique transverse intersection with the next. Each surface diagram co…

We explore Jaeger's state model for the HOMFLYPT polynomial. We reformulate this model in the language of Gauss diagrams and use it to obtain Gauss diagram formulas for a two-parameter family of Vassiliev invariants coming from the HOMFLYPT polynomial. These formulas are new already for invariants of degree 3.

This paper is an introduction to the state sum model for the Alexander-Conway polynomial that was introduced in the the author's book "Formal Knot Theory" (Princeton University Press, 1983). The article outlines how Alexander's original definition of the polynomial as the determinant of a matrix associated with the lin…

The abstract extends Reidemeister theorem to 3-manifolds using diagrams of links and bands.

A method for vectorizing persistence diagrams simplifies topological data analysis.

Given a diagram of a link K in S^3, we write down a Heegaard diagram for the branched-double cover Sigma(K). The generators of the associated Heegaard Floer chain complex correspond to Kauffman states of the link diagram. Using this model we make some computations of the homology \hat{HF}(Sigma(K)) as a graded group. W…

New proof shows knot Floer thickness limits bad domains in diagrams.

This paper studies posets associated with link diagrams and their algebraic properties.

In this paper, we discuss the (co)homology theory of biquandles, derived biquandle cocycle invariants for oriented surface-links using broken surface diagrams and how to compute the biquandle cocycle invariants from marked graph diagrams. We also develop the shadow (co)homology theory of biquandles and construct the sh…

This paper defines a new invariant of virtual knots and links that we call the extended bracket polynomial, and denote by <<K>> for a virtual knot or link K. This invariant is a state summation over bracket states of the oriented diagram for K. Each state is reduced to a virtual 4-regular graph in the plane and the pol…

We study a canonical spanning surface obtained from a knot or link diagram depending on a given Kauffman state, and give a sufficient condition for the surface to be essential. By using the essential surface, we can see the triviality and splittability of a knot or link from its diagrams. This has been done on the exte…

In [14], the second named author constructed the bracket invariant [.] of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams K, the following formula holds: [K]=K', where K' is the underlying graph of the diagram, i.e., the…

We work with a generalization of knot theory, in which one diagram is reachable from another via a finite sequence of moves if a fixed condition, regarding the existence of certain morphisms in an associated category, is satisfied for every move of the sequence. This conditional setting leads to a possibility of irreve…

Dye and Kauffman defined surface bracket polynomials for virtual links by use of surface states, and found a relationship between the surface states and the minimal genus of a surface in which a virtual link diagram is realized. They and Miyazawa independently defined a multivariable polynomial invariant of virtual lin…

We produce a facial state sum on plane diagrams of a knot or a link which admits an invariant specialization under Polyak's recent set of generating of 4 Reidemeister moves. Thus an isotopy invariant of framed links is obtained. Each state is a complete coloring of the faces of the diagram into white and black faces so…

A new framework for structured bandits using influence diagrams and variational Thompson sampling.

We present a new algorithm for exactly solving decision making problems represented as influence diagrams. We do not require the usual assumptions of no forgetting and regularity; this allows us to solve problems with simultaneous decisions and limited information. The algorithm is empirically shown to outperform a sta…

Paper derives explicit formulas for AJ-bracket of tied links.

New method converts complex PDs into stable vectors for ML.

We define polynomial tangle invariants $\nabla_T^s$ via Kauffman states and Alexander codes and investigate some of their properties. In particular, we prove symmetry relations for $\nabla_T^s$ of 4-ended tangles and deduce that the multivariable Alexander polynomial is invariant under Conway mutation. The invariants $…

Let $(M,ξ)$ be a contact 3-manifold. We present two new algorithms, the first of which converts an open book $(Σ,Φ)$ supporting $(M,ξ)$ with connected binding into a contact surgery diagram. The second turns a contact surgery diagram for $(M,ξ)$ into a supporting open book decomposition. These constructions lead to a r…

Bayesian method classifies actin cytoskeleton networks using topological data.

Let D be a link diagram with n crossings, s_A and s_B its extreme states and |s_AD| (resp. |s_BD|) the number of simple closed curves that appear when smoothing D according to s_A (resp. s_B). We give a general formula for the sum |s_AD|+|s_BD| for a k-almost alternating diagram D, for any k, characterising this sum as…

Differential Cohomotopy theory predicts brane interactions via chord diagrams.