Journal of Knot Theory and Its Ramifications

In this paper, we introduce the definition of nonseparating disk pants graph for a handlebody, and prove that for $H_{g,1}(g\ge 2)$, the nonseparating disk pants graph is connected.

This paper is concerned with the crosscap numbers of Montesinos knots. For a family of Montesinos knots, we give a lower bound of $-\chi /\#s$ among all essential surfaces $F$ embedded in the exterior, where $-\chi /\#s$ denotes the ratio of negative Euler characteristic of the surface and the number of sheets. From this result, we determine the crosscap numbers of a family of Montesinos knots. Our method relies on the algorithm of enumerating all essential surfaces for Montesinos knots given by Hatcher and Oertel.

The Slope Conjecture and the Strong Slope Conjecture predict that the degree of the colored Jones polynomial of a knot is matched by the boundary slope and the Euler characteristic of some essential surfaces in the knot complement. By solving a problem of quadratic integer programming to find the maximal degree and using the Hatcher–Oertel edgepath system to find the corresponding essential surface, we verify the Slope Conjectures for a family of 3-string Montesinos knots satisfying certain conditions.

In the paper [13], for an arbitrary virtual link $L$, three groups $G_{1,r}(L)$, $r>0$, $G_2(L)$ and $G_3(L)$ were defined. In the present paper, these groups for the virtual trefoil are investigated. The structure of these groups are found out and the fact that some of them are not isomorphic to each other is proved. Also, we prove that $G_3$ distinguishes the Kishino knot from the trivial knot. The fact that these groups have the lower central series which does not stabilize on the second term is noted. Hence, we have a possibility to study these groups using quotients by terms of the lower central series and to construct representations of these groups in rings of formal power series. It allows to construct an invariants for virtual knots.

We consider a compact hyperbolic antiprism. It is a convex polyhedron with $2n$ vertices in the hyperbolic space ${ℍ}^3$. This polyhedron has a symmetry group $S_{2n}$ generated by a mirror-rotational symmetry of order $2n$, i.e. rotation to the angle $\pi /n$ followed by a reflection. We establish necessary and sufficient conditions for the existence of such polyhedra in ${ℍ}^3$. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formulas expressing the volume of a hyperbolic antiprism in terms of the edge lengths.

In [1], the authors proved a sliceness criterion for odd free knots: free knots with odd chords. In the present paper, we give a similar criterion for stably odd free knots.

In essence, free knots may be considered as framed 4-graphs. That leads to an important notion of framed 4-graph cobordism and the associated notion of graph genera.

Some additional results on graph and free knot sliceness and cobordism are given.

We use Kirk’s invariant of link maps $S^2\bigsqcup S^2\to S^4$ and its variations due to Koschorke and Kirk–Livingston to deduce results about classical links. Namely, we give a new proof of the Nakanishi–Ohyama classification of two-component links in $S^3$ up to $\mathrm{\Delta }$-link homotopy. We also prove its version for string links, which is due (in a slightly different form) to Fleming–Yasuhara. The proofs do not use Clasper Theory.

We introduce a natural $G_n^k$-type construction of groups on any given simplicial complex $K$ called $G^k(K)$ as a generalization of the notion of groups $G_n^k$, which gives a functor from the category of simplicial complexes and injective simplicial maps to the category of groups. We prove that the word problem on $G^2(K)$ is solvable.

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.

We introduce two sequences of two-variable polynomials ${\{L_K^n(t,\ell )\}}_{n=1}^{\infty }$ and ${\{F_K^n(t,\ell )\}}_{n=1}^{\infty }$, expressed in terms of index value of a crossing and $n$-dwrithe value of a virtual knot $K$, where $t$ and $\ell$ are variables. Basing on the fact that $n$-dwrithe is a flat virtual knot invariant, we prove that $L_K^n$ and $F_K^n$ are virtual knot invariants containing Kauffman affine index polynomial as a particular case. Using $L_K^n$ we give sufficient conditions when virtual knot does not admit cosmetic crossing change.