Journal of Knot Theory and Its Ramifications

We construct a complete invariant of oriented connected closed surfaces in ${𝕊}^3$, which generalizes the notion of peripheral system of a knot group. As an application, we define two computable invariants to investigate handlebody knots and bi-knotted surfaces with homeomorphic complements. In particular, we obtain an alternative proof of inequivalence of Ishii, Kishimoto, Moriuchi and Suzuki’s handlebody knots $5_1$ and $6_4$.

We give a lower bound of the crosscap number of alternating knots using band surgery. This lower bound is equal to the minimum number of the first Betti number, i.e. one minus the maximal Euler characteristic, over all (orientable and non-orientable) surfaces spanning a given alternating knot.

In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These virtual graphs are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the $S$-polynomial, and formulate the $𝔰𝔩(N)$ Penrose polynomial for non-cubic graphs, giving contraction–deletion relations. The $S$-polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the $S$-polynomial at squares of integers.

We extend the classical definition of width to higher dimensional, smooth codimension two knots and show in each dimension there are knots of arbitrarily large width.

We give a short elementary proof that a Khovanov-type link homology constructed from a diagonalizable Frobenius algebra is degenerate.

For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an descending diagram. We study the $\mathrm{\Gamma }$-polynomial of knots with a given ascending number. We give a lower bound of the ascending numbers by using $\mathrm{\Gamma }$-polynomials. We estimate the ascending numbers for 65 prime knots up to 10 crossings by using $\mathrm{\Gamma }$-polynomials, Conway polynomials, and the determinants.

The group of planar (or flat) pure braids on $n$ strands, also known as the pure twin group, is the fundamental group of the configuration space $F_{n,3}(ℝ)$ of $n$ labeled points in $ℝ$ no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note, we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.

We consider embeddings of 3-manifolds in $S^4$ such that each of the two complementary regions has an abelian fundamental group. In particular, we show that an homology handle $M$ has such an embedding if and only if ${\pi }_1{(M)}^{\prime }$ is perfect, and that the embedding is then essentially unique.

This paper concerns the tree invariants of string links, introduced by Kravchenko and Polyak, which are closely related to the classical Milnor linking numbers also known as $\bar{\mu }$-invariants. We prove that, analogously as for $\bar{\mu }$-invariants, certain residue classes of tree invariants yield link-homotopy invariants of closed links. The proof is arrow diagramatic and provides a more geometric insight into the indeterminacy through certain tree stacking operations. Further, we show that the indeterminacy of tree invariants is consistent with the original Milnor’s indeterminacy. For practical purposes, we also provide a recursive procedure for computing arrow polynomials of tree invariants.

A point in the $(N,q)$-torus knot in ${ℝ}^3$ goes $q$ times along a vertical circle while this circle rotates $N$ times around the vertical axis. In the Lissajous-toric knot $K(N,q,p)$, the point goes along a vertical Lissajous curve (parametrized by $t\mapsto (sin(qt+\varphi ),cos(pt+\psi )))$ while this curve rotates $N$ times around the vertical axis. Such a knot has a natural braid representation $B_{N,q,p}$ which we investigate here. If $\mathrm{\text{gcd}}(q,p)=1$, $K(N,q,p)$ is ribbon; if $\mathrm{\text{gcd}}(q,p)=d>1$, $B_{N,q,p}$ is the $d$th power of a braid which closes in a ribbon knot. We give an upper bound for the $4$-genus of $K(N,q,p)$ in the spirit of the genus of torus knots; we also give examples of $K(N,q,p)$’s which are trivial knots.