Vassiliev Invariants Research Articles

Clasp pass moves and arrow polynomials of virtual knots

For classical knots, clasp pass moves are closely related to Vassiliev invariants of degree [Formula: see text]. Tsukamoto showed that the values of the Vassiliev invariant of degree [Formula: see text] induced from the Jones polynomial for two knots differ by [Formula: see text] or [Formula: see text], if they are related by a single clasp pass move. For virtual knots, the arrow polynomial is a generalization of the Jones polynomial and induces a Vassiliev invariant of degree [Formula: see text]. We show that the values of the Vassiliev invariant of degree [Formula: see text] induced from the arrow polynomial of two virtual knots differ by [Formula: see text] or [Formula: see text], if they are related by a single clasp pass move. We also obtain a lower bound of the distance between virtual knots by clasp pass moves.

Embedding calculus and grope cobordism of knots

We show that embedding calculus invariants evn are surjective for long knots in an arbitrary 3-manifold. This solves some remaining open cases of Goodwillie–Klein–Weiss connectivity estimates, and at the same time confirms one half of the conjecture that for classical knots evn are universal additive Vassiliev invariants over the integers. In addition, we give a sufficient condition for this conjecture to hold over a coefficient group, which is by recent results of Boavida de Brito and Horel fulfilled for the rationals and for the p-adic integers in a range. Therefore, embedding calculus invariants are strictly more powerful than the Kontsevich integral.Furthermore, our work shows they are more computable as well. Namely, the main theorem computes the first possibly non-vanishing invariantevnof a knot which is grope cobordant to the unknot to be precisely equal to the equivalence class of the underlying decorated tree of the grope in the associated graph complex. Actually, our techniques apply beyond dimension 3, offering a description of the layers in embedding calculus for long knots in a manifold of any dimension, and suggesting that certain generalised gropes realise the corresponding graph complex classes.

On the 3-loop polynomial of genus 1 knots with trivial Alexander polynomial

We give a restriction of the set of possible values of the 3-loop polynomials of genus 1 knots with trivial Alexander polynomial. As a special case, we present the 3-loop polynomial of any genus 1 knot with trivial [Formula: see text]-loop polynomials by using five Vassiliev invariants of the knot. Further, we give a new example of the calculation of the 3-loop polynomial.

Revisiting the second Vassiliev (In)variant for polymer knots

Knots in open strands such as ropes, fibers, and polymers, cannot typically be described in the language of knot theory, which characterizes only closed curves in space. Simulations of open knotted polymer chains, often parameterized to DNA, typically perform a closure operation and calculate the Alexander polynomial to assign a knot topology. This is limited in scenarios where the topology is less well-defined, for example when the chain is in the process of untying or is strongly confined. Here, we use a discretized version of the Second Vassiliev Invariant for open chains to analyze Langevin Dynamics simulations of untying and strongly confined polymer chains. We demonstrate that the Vassiliev parameter can accurately and efficiently characterize the knotted state of polymers, providing additional information not captured by a single-closure Alexander calculation. We discuss its relative strengths and weaknesses compared to standard techniques, and argue that it is a useful and powerful tool for analyzing polymer knot simulations.

On Vassiliev invariants of virtual knots

We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual knot invariants developed in [1]. In particular, following the theory of quantum invariants we work with ‘rotational’ virtual knots. We define chord diagrams, weight systems, and give examples of Lie algebra weight systems of rotational virtual knots. We end with a discussion of extended quantum invariants, which capture information that standard quantum invariants of rotational virtuals cannot.

A complete invariant for doodles on a 2-sphere

We define a complete invariant for doodles on a 2-sphere which takes values in series of chord diagrams of certain type. The coefficients at the diagrams with $n$ chords are finite type invariants of doodles of order at most $2n$.

On Finite Type Invariants of Welded String Links and Ribbon Tubes

Welded knotted objects are a combinatorial extension of knot theory, which can be used as a tool for studying ribbon surfaces in $4$-space. A finite type invariant theory for ribbon knotted surfaces was developed by Kanenobu, Habiro and Shima, and this paper proposes a study of these invariants, using welded objects. Specifically, we study welded string links up to $w_k$-equivalence, which is an equivalence relation introduced by Yasuhara and the second author in connection with finite type theory. In low degrees, we show that this relation characterizes the information contained by finite type invariants. We also study the algebraic structure of welded string links up to $w_k$-equivalence. All results have direct corollaries for ribbon knotted surfaces.

Gauss diagram formulae for Vassiliev invariants from Kauffman polynomial

A state model for Kauffman polynomial of Dubrovnik-version is described using Gauss diagrams. Based on the state model, the Gauss diagram formulae for Vassiliev invariants are given from the coefficients of Kauffman polynomial following the method of Chmutov and Polyak. Some arrow diagram identities are given to simplify the Gauss diagram formulae of order 3, which give Polyak–Viro and Chmutov–Polyak formulae for the Vassiliev invariant of order 3. The models of Kauffman polynomial and HOMFLY-PT polynomial give different Gauss diagram expressions when specializing in Jones polynomial.

Yarn ball knots and faster computations

We make use of the 3D nature of knots and links to find savings in computational complexity when computing knot invariants such as the linking number and, in general, most finite type invariants. These savings are achieved in comparison with the 2D approach to knots using knot diagrams.

Data science analysis of Vassiliev invariants and knot similarity based on distributed machine learning

INTRODUCTION: Knot theory has a long history, and as a branch of topology, it has received extensive attention. At present, the scientific analysis of data based on the similarity of Vassiliev invariants and knots under machine learning technology is the focus of the mathematical community. However, at present, there are some difficulties in the research work on the similarity of Vassiliev invariants and knots. These difficulties not only delay the progress of Vassiliev invariants research, but also slow down the speed of knot similarity research.OBJECTIVES: However, with the acceleration of the intelligent process, various intelligent technologies have been applied in the research of mathematics, biology and physics, providing excellent help for the research of many disciplines. Therefore, machine learning technology could be used to carry out new research on Vassiliev invariants and knot similarity.METHODS: Traditional knot analysis technology was combined with machine learning technology to find a more efficient and stable way of exploring Vassiliev invariants and knot similarity. his paper proposed a research method of data scientific analysis based on Vassiliev invariants and knot similarity under machine learning technology. Its purpose was to combine traditional knot research methods with machine learning technology to improve the efficiency of knot research. The algorithm proposed in this paper was the knot Vassiliev invariant analysis algorithm based on machine learning, which could use the intelligent and efficient analysis algorithm of machine learning technology to process the data of complex knots. This algorithm has improved the accuracy of the analysis of knot characteristics, and reduced the analysis time and the memory consumption at runtime.RESULTS: By testing the similarity between the Vassiliev invariant based on machine learning and the knot, the results showed that the analysis accuracy of the traditional Vassiliev invariant computing technology for the chiral characteristics, the number of intersections and the number of knots in the knot image was 84.25%, 83.27% and 85.56% respectively. The accuracy of knot Vassiliev invariant analysis algorithm based on machine learning for these indicators was 91.87%, 92.66% and 92.12% respectively. Obviously, the knot Vassiliev invariant analysis algorithm based on machine learning was superior to the traditional knot computing technology, and its analysis results were more excellent.CONCLUSION: In general, the research topic proposed in this paper has been proved to be of practical value. This research result proved that machine learning technology could play an excellent role in the current knot research, which correspondingly expanded the research direction of Vassiliev invariants and knot similarity.

The coefficients of the Jones polynomial

It has been known that the coefficients of the series expansion of the Jones polynomial evaluated at [Formula: see text] are rational-valued Vassiliev invariants. In this paper, we calculate minimal multiplying factor, [Formula: see text], needed for these rational-valued invariants to become integer-valued Vassiliev invariants. By doing that, we obtain a set of integer-valued Vassiliev invariants.

On constraints for knots to admit chirally cosmetic surgeries and their calculations

We discuss various constraints for knots in $S^{3}$ to admit chirally cosmetic surgeries, derived from invariants of 3-manifolds, such as, the quantum $SO(3)$-invariant, the rank of the Heegaard Floer homology, and finite type invariants. We apply them to show that a large portion (roughly 75$\%$) of knots which are neither amphicheiral nor $(2,p)$-torus knots with less than or equal to 10 crossings admits no chirally cosmetic surgeries.

A Kontsevich integral of order 1

We define a 1-cocycle in the space of long knots that is a natural generalization of the Kontsevich integral seen as a 0-cocycle. It involves a 2-form that generalizes the Knizhnik--Zamolodchikov connection. We show that the well-known close relationship between the Kontsevich integral and Vassiliev invariants (via the algebra of chord diagrams and 1T-4T relations) is preserved between our integral and Vassiliev 1-cocycles, via a change of variable similar to the one that led Birman--Lin to discover the 4T relations. We explain how this construction is related to Cirio--Faria Martins' categorification of the Knizhnik--Zamolodchikov connection.

VARIATIONS OF MILNOR’S TRIPLE LINKING NUMBER

Topological polymers have various topological types, and they are expressed by graphs. However, the Jones polynomial, we have a difficulty to compute it; computational time is growing exponentially with respect to the crossing number. The simplest Vassiliev invariant is the linking number and thus we will seek a next simple one is as the Milnor's triple linking number. In this paper, we introduce simple Gauss diagram formulas of Vassiliev invariants of Milnor type. These are non-torsion valued, whereas the base-point-free Milnor's triple linking number is usually torsion-valued.

Asymptotics and statistics on Fishburn matrices: Dimension distribution and a conjecture of Stoimenow

AbstractWe establish the asymptotic normality of the dimension of large‐size random Fishburn matrices by a complex‐analytic approach. The corresponding dual problem of size distribution under large dimension is also addressed and follows a quadratic type normal limit law. These results represent the first of their kind and solve two open questions raised in the combinatorial literature. They are presented in a general framework where the entries of the Fishburn matrices are not limited to or nonnegative integers . The analytic saddle‐point approach we apply, based on a powerful transformation for ‐series due to Andrews and Jelínek, is also useful in solving a conjecture of Stoimenow in Vassiliev invariants.

Goussarov–Polyak–Viro conjecture for degree three case

Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas has been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to present the seven invariants of the degree three (Proposition 4). We further give [Formula: see text] Gauss diagram formulas of classical knots (Proposition 5). In particular, the Polyak–Viro Gauss diagram formula [M. Polyak and O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not. 1994 (1994) 445–453] is not a long virtual knot invariant; however, it is included in the list of [Formula: see text] formulas. It has been unknown whether this formula would be available by arrow diagram calculus automatically. In consequence, as it relates to the conjecture of Goussarov-Polyak-Viro [Finite-type invariants of classical and virtual knots, Topology 39 (2000) 1045–1068, Conjecture 3.C], for all the degree three finite type long virtual knot invariants, each Gauss diagram formula is represented as those of Vassiliev invariants of classical knots (Theorem 1).

Milnor invariants of sorting networks

In this note we investigate finite-type invariants of the pure braids arising from signed sorting networks.

Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure

We have recently proposed [1] a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with SU(N) gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values.First, we discuss the computation of Vassiliev invariants. Second, we discuss the Vogel theorem of not distinguishing chord diagrams by weight systems coming from semisimple Lie (super)algebras. Third, we provide a method for constructing linear recursive relations for the colored Jones polynomials considering a special case of torus knots T[2,2k+1]. Fourth, we give a generalization of the one-hook scaling property for the colored Alexander polynomials. And finally, for the group factors we provide a combinatorial description, which has a clear dependence on the rank N and the representation R.

Two-bridge knots admit no purely cosmetic surgeries

We show that two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries, i.e., no pair of distinct Dehn surgeries on such a knot produce 3-manifolds that are homeomorphic as oriented manifolds. Our argument, based on a recent result by Hanselman, uses several invariants of knots or 3-manifolds; for knots, we study the signature and some finite type invariants, and for 3-manifolds, we deploy the $SL(2,\mathbb{C})$ Casson invariant.

Chern-Simons perturbative series revisited

A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with SU(N) gauge group is studied in symmetric approach. A special basis in the center of the universal enveloping algebra ZU(slN) is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods have wide applications, the most straightforward and evident ones are mentioned. Namely, Vassiliev invariants of higher orders are computed, a conjecture about existence of new symmetries of the colored HOMFLY polynomials is stated, and the recently discovered tug-the-hook symmetry of the colored HOMFLY polynomial is proved.