Journal of Knot Theory and Its Ramifications

We compute the bridge spectra of cables of 2-bridge knots. We also give some results about bridge spectra and distance of Montesinos knots.

For coprime integers $p(>0)$ and $q$, the $(p,q)$-cable $\mathrm{\Gamma }$-polynomial of a knot is the $\mathrm{\Gamma }$-polynomial of the $(p,q)$-cable knot of the knot, where the $\mathrm{\Gamma }$-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. In this paper, we show that there exist infinitely many knots with the trivial $(2,1)$-cable $\mathrm{\Gamma }$-polynomial, that is, the $(2,1)$-cable $\mathrm{\Gamma }$-polynomial of the trivial knot. Moreover, we see that the knots have the trivial $\mathrm{\Gamma }$-polynomial, the trivial first coefficient HOMFLYPT and Kauffman polynomials.

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.

Let $M$ be a non-abelian semi-direct product of a cyclic group $\mathbb{Z}/n$ and an elementary abelian $p$-group $A=\oplus k(\mathbb{Z}/p)$ of order $p^k$, $p$ being a prime and $gcd(n,p)=1$. Suppose that the knot group $G(K)$ of a knot $K$ in the $3$-sphere is represented on $M$. Then we conjectured (and later proved) that the twisted Alexander polynomial ${\mathrm{\Delta }}_{\gamma ,K}(t)$ associated to $\gamma :G(K)\to M\to GL(p^k,\mathbb{Z})$ is of the form: $\frac{{\mathrm{\Delta }}_K(t)}{1-t}F(t^n)$, where ${\mathrm{\Delta }}_K(t)$ is the Alexander polynomial of $K$ and $F(t^n)$ is an integer polynomial in $t^n$. In this paper, we present a proof of the following. For a $2$-bridge knot $K(r)$ in $H(p)$, if $n=2$ and $k=1$, then $F(t^2)$ is written as $f(t)f(t^{-1})$, where $H(p)$ is the set of $2$-bridge knots whose knot groups map on that of $K(1/p)$ with $p$ odd.

In this paper, we consider generalizations of the Alexander polynomial and signature of 2-bridge knots by considering the Gordon–Litherland bilinear forms associated with essential state surfaces of the 2-bridge knots. We show that the resulting invariants are well-defined and explore properties of these invariants. Finally, we realize the boundary slopes of essential surfaces as differences of signatures of the knot.

Given any oriented link diagram, one can construct knot invariants using skein relations. Usually such a skein relation contains three or four terms. In this paper, the author introduces several new ways to smooth a crossings, and uses a system of skein equations to construct link invariants. This invariant can also be modified by writhe to get a more powerful invariant. The modified invariant is a generalization of both the HOMFLYPT polynomial and the two-variable Kauffman polynomial. Using the diamond lemma, a simplified version of the modified invariant is given. It is easy to compute and is a generalization of the two-variable Kauffman polynomial.

It is known that there are 21 ribbon knots with 10 crossings or fewer. We show that for every ribbon knot, there exists a tangle that satisfies two properties associated with the knot. First, under a specific closure, the closed tangle is equivalent to its corresponding knot. Second, under a different closure, the closed tangle is equivalent to the unlink. For each of these 21 ribbon knots, we present a 4-strand tangle that satisfies these properties. We provide diagrams of these tangles and also express them in planar diagram notation.

We introduce a new topological invariant $\nu (X)\in \mathbb{N}\cup \{0\}$ of compact manifolds-with-boundaries $X$ which is much connected with boundary-unions. A boundary-union is a kind of decomposition of compact manifolds-with-boundaries. See the body of the paper for the precise definition. Let $M$ and $N$ be $m$-dimensional compact manifolds-with-boundaries. Let $M{\cup }_{\partial }N$ be a boundary-union of $M$ and $N$. Then we have

We show that quandle coverings in the sense of Eisermann form a (regular epi)-reflective subcategory of the category of surjective quandle homomorphisms, both by using arguments coming from categorical Galois theory and by constructing concretely a centralization congruence. Moreover, we show that a similar result holds for normal quandle extensions.

We present a marked analog of Carter and Saito’s movie theorem. Our definition of marking was chosen to coincide with the markings that arise in link Floer homology. In order to deal with complications arising from certain isotopes, we define three equivalences for marked surfaces and work over an equivalence class of marked surfaces when proving our generalization of Carter and Saito’s movie theorem.

A quadrisecant line of a knot $K$ is a straight line which intersects $K$ in four points, and a quadrisecant is a 4-tuple of points of $K$ which lie in order along the quadrisecant line. If $K$ has a finite number of quadrisecants, take $W$ to be the set of points of $K$ which are in a quadrisecant. Replace each subarc of $K$ between two adjacent points of $W$ along $K$ with the straight line segment between them. This gives the quadrisecant approximation of $K$. It was conjectured that the quadrisecant approximation is always a knot with the same knot type as the original knot. We show that every knot type contains two knots, the quadrisecant approximation of one knot has self-intersections while the quadrisecant approximation of the other knot is a knot with a different knot type.