Journal of Knot Theory and Its Ramifications

In this paper, we study the Gordian metric on the set of all theta-curves and give a lower bound of it. We define the Gordian complex of theta-curves, which is a simplicial complex whose vertices consist of all theta-curves in the 3-dimensional Euclidean space ${ℝ}^3$. We show that for any given theta-curve $\mathrm{\Theta }$, there exists an infinite family of theta-curves containing $\mathrm{\Theta }$ such that the Gordian distance between any two distinct members of this family is equal to one.

We discuss links in thickened surfaces. We define the Khovanov–Lipshitz–Sarkar stable homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened surfaces with genus $>1$. A surface means a closed oriented surface unless otherwise stated. Of course, a surface may or may not be the sphere. A thickened surface means a product manifold of a surface and the interval. A link in a thickened surface (respectively, a 3-manifold) means a submanifold of a thickened surface (respectively, a 3-manifold) which is diffeomorphic to a disjoint collection of circles. Our Khovanov–Lipshitz–Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus $>1$ are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus $>1$. It is the first meaningful Khovanov–Lipshitz–Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. We point out that our theory has a different feature in the torus case.

We recall a skein relation of the first coefficient Kauffman polynomial for knots. By using the skein relation, we show that there exist infinitely many knots whose Whitehead doubles have the trivial first coefficient Kauffman polynomial.

For a knot $K$, the doubly slice genus $g_{ds}(K)$ is the minimal $g$ such that $K$ divides a closed, orientable, and unknotted surface of genus $g$ embedded in $S^4$. In this paper, we identify the doubly slice genera of 2909 of the 2977 prime knots which have a crossing number of 12 or fewer.

In this paper, we show that there exists a knot which has infinitely many non-equivalent symmetric union presentations with the same partial knots. We also define an invariant of a symmetric union presentation and give alternative proofs to results of Eisermann and Lamm and a result of Collari and Lisca by using methods in 3-dimensional topology.

In this paper, we define the virtual Alexander polynomial following the works of Boden et al. (2016) [Alexander invariants for virtual knots, J. Knot Theory Ramications 24(3) (2015) 1550009] and Kaestner and Kauffman [Parity biquandles, in Knots in Poland. III. Part 1, Banach Center Publications, Vol. 100 (Polish Academy of Science Mathematical Institute, Warsaw, 2014), pp. 131–151]. The properties of this invariant are explored and some examples are computed. In particular, the invariant demonstrates that many virtual knots cannot be unknotted by crossing changes on only odd crossings.

Generalized Montesinos tangles are classified, and the system of unoriented tangle equations $N(U+P)=K_1$ and $N(U+R)=K_2$ is solved for a generalized Montesinos tangle $U$ where $P$ and $R$ are rational tangles and $K_1$ and $K_2$ are Montesinos knots/links.

We show that the equivariant and non-equivariant non-orientable 4-genus of $p$-periodic knots may differ, for any choice of $p\ge 2$. Similar results have previously been obtained for the smooth 4-genus and non-orientable 3-genus of a periodic knot. These stand in contrast to Edmonds’ acclaimed result by which the equivariant and non-equivariant Seifert genus of a periodic knot agree.

We derive a minimal generating set of planar moves for diagrams of surfaces embedded in the four-space. These diagrams appear as the bonded classical unlink diagrams.

By the Tait Flyping Conjecture, all possible minimal diagrams of an alternating knot are related by a finite sequence of flypes. We use this fact for an algorithm to produce all possible minimal diagrams of an oriented alternating prime knot type. Starting with any given oriented minimal alternating diagram of a prime knot, the algorithm performs flypes exhaustively until all other possible minimal diagrams of the oriented knot type have been produced. Results for all distinct alternating prime knot types up to 16 crossings are shown. In addition, prime knot families are discussed that have either a small or a very large number of distinct minimal diagrams.