Finite-type invariants of classical and virtual knots

For ordinary knots in R3, there are no degree one Vassiliev invariants. For virtual knots, however, the space of degree one Vassiliev invariants is infinite dimensional. We introduce a sequence of three degree one Vassiliev invariants of virtual knots of increasing strength. We demonstrate that the strongest invariant is a universal Vassiliev invariant of degree one for virtual knots in the sense that any other degree one Vassiliev invariant can be recovered from it by a certain natural construction. To prove these results, we extend the based matrix invariant introduced by Turaev for virtual strings to the class of singular virtual knots with one double-point.

We introduce a four-variable index polynomial invariant of long virtual knots that is non-trivial for many long virtual knots, but is trivial for classical knots so that it is an extension of a one-variable index polynomial invariant introduced in [A polynomial invariant of long virtual knots, European J. Combin.30 (2009) 1289–1296]. We give various properties of this polynomial and examples. Also, we use this polynomial invariant for long virtual knots to distinguish virtual knots, and we obtain a polynomial invariant for long flat virtual knots which is very useful to determine whether long flat virtual knots are invertible or not.

Milnor's $\bar{\mu}$-invariants of links in the $3$-sphere $S^3$ vanish on any link concordant to a boundary link. In particular, they are trivial on any knot in $S^3$. Here we consider knots in thickened surfaces $\Sigma \times [0,1]$, where $\Sigma$ is closed and oriented. We construct new concordance invariants by adapting the Chen-Milnor theory of links in $S^3$ to an extension of the group of a virtual knot. A key ingredient is the Bar-Natan $\textit{Zh}$ map, which allows for a geometric interpretation of the group extension. The group extension itself was originally defined by Silver-Williams. Our extended $\bar{\mu}$-invariants obstruct concordance to homologically trivial knots in thickened surfaces. We use them to give new examples of non-slice virtual knots having trivial Rasmussen invariant, graded genus, affine index (or writhe) polynomial, and generalized Alexander polynomial. Furthermore, we complete the slice status classification of all virtual knots up to five classical crossings and reduce to four (out of 92800) the number of virtual knots up to six classical crossings having unknown slice status. Our main application is to Turaev's concordance group $\mathscr{VC}$ of long knots on surfaces. Boden and Nagel proved that the concordance group $\mathscr{C}$ of classical knots in $S^3$ embeds into the center of $\mathscr{VC}$. In contrast to the classical knot concordance group, we show $\mathscr{VC}$ is not abelian; answering a question posed by Turaev.

We introduce a new technique for studying classical knots with the methods of virtual knot theory. Let K be a knot and J be a knot in the complement of K with lk (J, K) = 0. Suppose there is covering space [Formula: see text], where V(J) is a regular neighborhood of J satisfying V(J) ∩ im (K) = ∅ and Σ is a connected compact orientable 2-manifold. Let K′ be a knot in Σ × (0, 1) such that πJ(K′) = K. Then K′ stabilizes to a virtual knot [Formula: see text], called a virtual cover of K relative to J. We investigate what can be said about a classical knot from its virtual covers in the case that J is a fibered knot. Several examples and applications to classical knots are presented. A basic theory of virtual covers is established.

Satoh and Taniguchi introduced the [Formula: see text]-writhe [Formula: see text] for each non-zero integer [Formula: see text], which is an integer invariant for virtual knots. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. In our previous paper, we showed that if [Formula: see text] is a sequence of integers with [Formula: see text], then there exists a virtual knot [Formula: see text] whose virtual unknotting number equals one and [Formula: see text] for any [Formula: see text]. In this paper, we show that there exist infinitely many virtual knots having such properties by using the vertex connected sum on Gauss diagrams.

Although the connected sum of trivial knots is trivial in classical knot theory, there is a nontrivial connected sum of trivial knots in virtual knot theory. In this paper, we give connected sum formulae of the intersection polynomials of virtual knots introduced in the preceding paper [6]. As an application, we show that there are infinitely many connected sums of any pair of virtual knots.

In this paper the classification of n-dimensional knots in S"+2, bounding /--connected manifolds, where 3r>n + l>6, in terms of stable homotopy theory is suggested.The problem of isotopy classification is the fundamental one of knot theory.It was solved by A. Haefliger and J. Levine for knots of codimension > 3. The first step in reduction of a classification of knots of codimension two to a homotopy problem was made by R. Lashof and J. Shaneson [6].They showed that in the class of -dimensional knots with group Z for n > 5 there are at most two different knots having homotopy equivalent exteriors.In 1970 J. Levine [10] gave an algebraic classification of (2q -l)-dimensional knots in S2q+X bounding (q -1)connected manifolds.It turned out that the only invariant determining the isotopy type of such knots is the Seifert matrix considered to within 5-equivalence.Later Trotter [16] and C. Kearton [5] obtained a classification of knots, studied by J. Levine, in terms of Blanchfield pairing.C. Kearton has partially analyzed the more difficult problem of classification of 2^r-dimensional knots in S2q+2 bounding (q -l)-connected manifolds.In the present paper the classification of a wider class of higher-dimensional knots is obtained.It is the class of -dimensional knots in S"+2 bounding r-connected manifolds, where 3r > n + 1 > 6.The main result of this paper is the construction of a one-to-one correspondence between the set of isotopy types of such knots and some set described in purely homotopic terms.The plan of the paper is as follows:In 1 submanifolds of the sphere of codimension one are considered.Such a submanifold is assigned some pairing in the sense of the Spanier-Whitehead theory.This pairing describes the homotopy linking of the submanifold with its copy translated in the direction of the positive normal field.This pairing induces the usual Seifert pairing on middle dimensional homology groups and is called the homotopy Seifert pairing.The main result of 1 is the construction of a one-to-one correspondence between the set of isotopy types of such submanifolds and the set of homotopy pairings.

Grope cobordism of classical knots

We study generalizations of finite-type knot invariants obtained by replacing the crossing change in the Vassiliev skein relation by some other local move, analyzing in detail the band-pass and doubled-delta moves. Using braid-theoretic techniques, we show that, for a large class of local moves, generalized Goussarov's n-equivalence classes of knots form groups under connected sum. (Similar results, but with a different approach, have been obtained before by Taniyama and Yasuhara.) It turns out that primitive band-pass finite-type invariants essentially coincide with standard primitive finite-type invariants, but things are more interesting for the doubled-delta move. The complete degree 0 doubled-delta invariant is the S-equivalence class of the knot. In this context, we generalize a result of Murakami and Ohtsuki to show that the only primitive Vassiliev invariants of S-equivalence taking values in an abelian group with no 2-torsion arise from the Alexander–Conway polynomial. We start analyzing degree one doubled-delta invariants by considering which Vassiliev invariants are of doubled-delta degree one, finding that there is exactly one such invariant in each odd Vassiliev degree, and at most one (which is ℤ2-valued) in each even Vassiliev degree. Analyzing higher doubled-delta degrees, we observe that the Euler degree n + 1 part of Garoufalidis and Kricker's rational lift of the Kontsevich integral is a doubled-delta degree 2n invariant.

In his two pioneering articles [9, 10] Jerry Levine introduced and completely determined the algebraic concordance groups of odd dimensional knots. He did so by defining a host of invariants of algebraic concordance which he showed were a complete set of invariants. While being very powerful, these invariants are in practice often hard to determine, especially for knots with Alexander polynomials of high degree. We thus propose the study of a weaker set of invariants of algebraic concordance—the rational Witt classes of knots. Though these are rather weaker invariants than those defined by Levine, they have the advantage of lending themselves to quite manageable computability. We illustrate this point by computing the rational Witt classes of all pretzel knots. We give many examples and provide applications to obstructing sliceness for pretzel knots. Also, we obtain explicit formulae for the determinants and signatures of all pretzel knots. This article is dedicated to Jerry Levine and his lasting mathematical legacy; on the occasion of the conference “Fifty years since Milnor and Fox” held at Brandeis University on June 2–5, 2008.

A method of representing knots, links, and singular knots and links by words in a finite alphabet and so-called d-diagrams is given. A representation of the chord diagram algebra by words in a finite alphabet is considered. This method, unlike coding by Gauss diagrams, allows one to consider knots and links simultaneously. An algorithm for recognition of diagrams of classical knots in terms of d-diagrams is given. Bibliography: 9 titles.

This paper introduces a self-linking invariant for virtual knots and links, and relates this invariant to a state model called the binary bracket, and to a class of coloring problems for knots and links that include classical coloring problems for cubic g

Kauffman and Lomonaco (New J Phys 4:73.1–73.18, 2002. arXiv:quant-ph/0401090, New J Phys 6:134.1–134.40, 2004) explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In Alagic et al. (Yang–Baxter operators need quantum entanglement to distinguish knots, 2015. arXiv:1507.05979v1), it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang–Baxter equation. We show that the arguments used by Alagic et al. (2015) generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang–Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of Alagic et al. (2015). We also give a class of R matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give an example of an SU(2) representation of the three-strand braid group that models the Jones polynomial for closures of three-strand braids. This invariant is a quantum model for the Jones polynomial restricted to three-strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing. The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement and the ER = EPR hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, to construct a new topological space that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.

Two definitions of the bridge index of a welded knot

There are some phenomena arising in the virtual knot theory which are not the case for classical knots. One of them deals with the "breaking" procedure of knots and obtaining long knots. Unlike the classical case, they might not be the same. The present work is devoted to construction of some invariants of long virtual links. Several explicit examples are given. For instance, we show how to prove the non-triviality of some knots obtained by breaking virtual unknot diagrams by very simple means.