Narrow Results

We construct new invariant polynomials in two and multiple variables for virtual knots and links. They are defined as determinants of Alexander-like matrices whose determinants are virtual link invariants. These polynomials vanish on classical links. In some cases, they separate links that can not be separated by the Jones–Kauffman polynomial [Kau] and the polynomial proposed in [Ma3].

We construct a multi-variable polynomial invariant for virtual knots and links via the concept of a decorated virtual magnetic graph diagram. The invariant is a generalization of the Jones–Kauffman polynomial for virtual knots and links. We show some features of the invariant including an evaluation of the virtual crossing number of a virtual knot or link.

We construct a multi-variable polynomial invariant Y for unoriented virtual links as a certain weighted sum of polynomials, which are derived from virtual magnetic graphs with oriented vertices, on oriented virtual links associated with a given virtual link. We show some features of the Y-polynomial including an evaluation of the virtual crossing number of a virtual link.

We construct a multi-variable polynomial invariant for virtual knots and links by using the concepts of a decorated virtual magnetic graph diagram and a weight map. We show that the invariant is a variety of virtual link polynomial with multiple variables introduced in [16] by the author and it gives a sharpened evaluation of the virtual crossing number of a virtual knot or link.

We introduce a polynomial invariant of virtual links that is non-trivial for many virtuals, but is trivial on classical links. Also this polynomial is sometimes useful to find the virtual crossing number of virtual knots. We give various properties of this polynomial and examples.

We introduce invariants of flat virtual links which are induced from Vassiliev invariants of degree one for virtual links. Also we give several properties of these invariants for flat virtual links and examples. In particular, if the value of some invariants of flat virtual knots F are non-zero, then F is non-invertible so that every virtual knot overlying F is non-invertible.

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.

We list prime knots with up to 12 crossings having the property: the lower bound of the unknotting number cannot be decided by the signature or non-triviality but is given by either: (i) the condition of a 2-bridge knot with unknotting number one, (ii) the criteria using special values of the Jones, Q, or HOMFLYPT polynomials, or (iii) Wendt's formula. Then we can give new information to the table of unknotting numbers in "KnotInfo", a web-based table of knot invariants.

In this paper, we describe a method of making a polynomial invariant of flat virtual knots in terms of an integer labeling of the flat virtual knot diagram and an invariant of virtual links. We show that the polynomial is sometimes useful to detect non-invertibility and also to determine the virtual crossing number of a given flat virtual knot.

In 1987, Przytyski and Traczyk introduced an algebraic structure, called a Conway algebra, and constructed an invariant of oriented links, which is a generalization of the HOMFLY-PT polynomial invariant. In 2018, Kim generalized a Conway algebra, which is an algebraic structure with two skein relations, which is called a generalized Conway algebra. In 2017, Joung, Kamada, Kawauchi and Lee constructed a polynomial invariant of oriented surface-links by using marked graph diagrams.

In this paper, we will introduce generalizations $MA$ and $\widehat{MA}$ of a Conway algebra and a generalized Conway algebra, which are called a marked Conway algebra and a generalized marked Conway algebra, respectively. We will construct invariants valued in $MA$ and $\widehat{MA}$ for oriented marked graphs and oriented surface-links by applying binary operations to classical crossings and marked vertices via marked graph diagrams. The polynomial invariant of oriented surface-links is obtained from the invariant valued in the marked Conway algebra with additional conditions.

In [Przytyski and Traczyk, Invariants of links of Conway type, Kobe J. Math.4 (1989) 115–139], Przytyski and Traczyk introduced an algebraic structure, called a Conway algebra, and constructed an invariant of oriented links, which is a generalization of the Homflypt polynomial invariant. On the other hand, in [Kauffman and Lambropoulou, New invariants of links and their state sum models, arXiv:1703.03655v2 [math.GT] 15 Mar 2017], Kauffman and Lambropoulou introduced new 4-variable invariants of oriented links, which are obtained by two computational steps: in the first step, we apply a skein relation on every mixed crossing to produce unions of unlinked knots. In the second step, we apply another skein relation on crossings of the unions of unlinked knots, which introduces a new variable. In this paper, we will introduce a generalization of the Conway algebra $Â$ with two binary operations and we construct an invariant valued in $Â$ by applying those two binary operations to mixed crossings and pure crossing, respectively. The 4-variable invariant of Kauffman and Lambropoulou with a specific condition is derived from the invariant valued in $Â$. Moreover, the generalized Conway algebra gives us an invariant of oriented links, which satisfies nonlinear skein relations.

This paper studies 2-adjacency between a 3-strand pretzel link and one of the Hopf link, the Solomon’s link and the Whitehead link by using the results that have been obtained about 2-adjacency between knots or links and their polynomials and etc. This paper shows that of all 3-strand pretzel links, only ordinary pretzel links are 2-adjacent to the Hopf link or the Solomon’s link or the Whitehead link. Conversely, these special links are not 2-adjacent to any other 3-strand pretzel links, except for themselves, respectively.

A multi-variable polynomial invariant for knotoids and linkoids, which is an enhancement of the bracket polynomial for knotoids introduced by Turaev, is given by using the concept of a pole diagram which originates in constructing a virtual link invariant. Several features of the polynomial are revealed.

We study 2-adjacency between a classical (3-strand) pretzel knot and the trefoil knot or the figure-eight knot by using the early results about classical pretzel knots and their polynomials and elementary number theory. We show that except for the trefoil knot or the figure-eight knot, a nontrivial classical pretzel knot is not 2-adjacent to either of them, and vice versa.

In this paper, we introduce the Goeritz matrix for a pseudo-link whose entries lie in the Laurent polynomial ring $\mathbb{Z}[u^{-1},u]$, which generalizes the Goeritz matrix for a classical link. We show that the determinant of a modified Goeritz matrix gives a Laurent polynomial invariant for pseudo-links in one variable u with integer coefficients. We also introduce the notions of the signature, determinant, and nullity of pseudo-links. Further, we discuss some properties of the invariants and compute the polynomials and those numerical invariants for several pseudo-knot families.

We construct a three-variable polynomial invariant for unoriented kd($1$)-linkoid diagrams including knotoid ones as a certain weighted sum of polynomials on oriented kd($1$)-linkoid diagrams associated with a given kd($1$)-linkoid diagram.

In this paper, we construct a polynomial invariant of Kauffman type for kd($1$)-linkoids and compute the polynomials for knotoids with up to three crossings. As a consequence, it is shown that the polynomial is different from the previous one in [Y. Miyazawa, A polynomial invariant of Kauffman type for knotoids, preprint].

In this paper, we introduce two new polynomial invariants $Q_D(t)$ and $A_D(t)$ for one-component virtual doodles. We will also show that these polynomial invariants are not invariants of flat virtual knots.

As a generalization of the classical knots, knotoids deal with the open ended knot diagram in a surface. In recent years, many polynomial invariants for knotoids appeared, such as the bracket polynomial, the index polynomial and the $n$th polynomial. In this paper, we introduce a new polynomial invariant $F$-polynomial for knotoids and discuss some properties of the $F$-polynomial. Finally, we construct a family of knotoid diagrams which can be distinguished from each other by the $F$-polynomial but cannot be distinguished by the index polynomial and the $n$th polynomial.

In this paper we extend the list of three manifolds for which the (2, ∞)-skein module is known by giving the first explicit calculations for non-trivial knot exteriors. We show that for the complement of a (2, 2p+1) torus knot the module is free with a very simple basis. As a consequence, we obtain a family of polynomial invariants for links in these manifolds. The invariants are analogous to the Jones polynomial for links in S³.