The intersection polynomials of a virtual knot II: Connected sums

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 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.

Finite-type invariants of classical and virtual knots

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.

We claim that HOMFLY polynomials for virtual knots, defined with the help of the matrix-model recursion relations, contain more parameters, than just the usual q and A=qN. These parameters preserve topological invariance and do not show up in the case of ordinary (non-virtual) knots and links. They are most conveniently observed in the hypercube formalism: then they substitute q-dimensions of certain fat graphs, which are not constrained by recursion and can be chosen arbitrarily. The number of these new topological invariants seems to grow fast with the number of non-virtual crossings: 0, 1, 1, 5, 15, 91, 784, 9160, ... This number can be decreased by imposing the factorization requirement for composites, in addition to topological invariance – still freedom remains. None of these new parameters, however, appears in HOMFLY for Kishino unknot, which thus remains unseparated from the ordinary unknots even by this enriched set of knot invariants.

We show that any virtual knot can be deformed into a trivial knot by a finite sequence of generalized Reidemeister moves and "forbidden moves". We also show that a Δ-unknotting operation is realized by a finite sequence of generalized Reidemeister moves and "forbidden moves".

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.

Study of stable isotopy classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces is considered as a planar analogue of virtual knot theory, a far reaching generalisation of classical knot theory. Recent works have established Alexander and Markov theorems in the planar setting. In the classical case, the role of groups is played by twin groups, a class of right-angled Coxeter groups. A new class of groups called virtual twin groups, that extends twin groups in a natural way, plays the role of groups in the virtual case. The virtual twin group V Tn contains the pure virtual twin group P V Tn, a planar analogue of the pure Artin braid group. In this paper, we prove that the pure virtual twin group P V Tn is an irreducible right-angled Artin group with trivial center and give it's precise presentation. We show that P V Tn has a decomposition as an iterated semi-direct product of infinite rank free groups. We give a complete description of the automorphism group of P V Tn and establish splitting of natural exact sequences of automorphism groups. As applications, we show that V Tn is residually finite and P V Tn has the R∞-property.

In this paper, we present the Goeritz matrix for checkerboard colorable virtual links or, equivalently, checkerboard colorable links in thickened surfaces Sg × [0, 1], which is an extension of the Goeritz matrix for classical knots and links in ℝ3. Using this, we show that the signature, nullity and determinant of classical oriented knots and links extend to those of checkerboard colorable oriented virtual links.

The 3-loop invariant (or, the 3-loop polynomial) of a knot is a rational form (or, a polynomial) presenting the 3-loop part of the Kontsevich invariant of knots. In this paper, we calculate the 3-loop polynomial of knots obtained by plumbing the doubles of two knots; this class of knots includes untwisted Whitehead doubles. We construct the 3-loop invariant by calculating the rational version of the Aarhus integral of a surgery presentation. As a consequence, we obtain an explicit presentation of the 3-loop polynomial for the knots.

We construct a new invariant polynomial for long virtual knots. It is a generalization of the Alexander polynomial. We designate it as ζ by meaning an analogy with ζ-polynomial for virtual links. The degree of ζ-polynomial estimates the virtual crossing number. We describe some application of ζ-polynomial for the study of minimal long virtual diagrams with respect to the number of virtual crossings.

A two-variable polynomial of virtual knots via 1-smoothing