Journal of Knot Theory and Its Ramifications

Motivated by problem of determining the unknotted routes for the scaffolding strand in DNA origami self-assembly, we examine the existence and knottedness of A-trails in graphs embedded on the torus. We show that any A-trail in a checkerboard-colorable torus graph is unknotted and characterizes the existence of A-trails in checkerboard-colorable torus graphs in terms of pairs of quasitrees in associated embeddings. Surface meshes are frequent targets for DNA nanostructure self-assembly, and so we study both triangular and rectangular torus grids. We show that aside from one exceptional family, a triangular torus grid contains an A-trail if and only if it has an odd number of vertices, and that such an A-trail is necessarily unknotted. On the other hand, while every rectangular torus grid contains an unknotted A-trail, we also show that any torus knot can be realized as an A-trail in some rectangular grid. Lastly, we use a gluing operation to construct infinite families of triangular and rectangular grids containing unknotted A-trails on surfaces of arbitrary genus. We also give infinite families of triangular grids containing no unknotted A-trail on surfaces of arbitrary nonzero genus.

Reshetikhin–Turaev (a.k.a. Chern–Simons) TQFT is a functor that associates vector spaces to two-dimensional genus $g$ surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a Macdonald $q,t$-deformation — refinement — of these operators that preserves the defining relations of the mapping class groups beyond genus 1. For this, we explicitly construct the refined TQFT representation of the genus 2 mapping class group in the case of rank one TQFT. This is a direct generalization of the original genus 1 construction of arXiv:1105.5117 opening a question that if it extends to any genus. Our construction is built upon a $q,t$-deformation of the square of $q$-6$j$ symbol of $U_q({\mathrm{\text{sl}}}_2)$, which we define using the Macdonald version of Fourier duality. This allows to compute the refined Jones polynomial for arbitrary knots in genus 2. In contrast with genus 1, the refined Jones polynomial in genus 2 does not appear to agree with the Poincare polynomial of the triply graded HOMFLY knot homology.

A spatial surface is a compact surface embedded in the $3$-sphere. In this paper, we provide several typical examples of spatial surfaces and construct a coloring invariant to distinguish them. The coloring is defined by using a multiple group rack, which is a rack version of a multiple conjugation quandle.

Letting $C$ be a compact $C^{\omega }$-curve embedded in the Euclidean $3$-space ($C^{\omega }$ means real analyticity), we consider a $C^{\omega }$-cuspidal edge $f$ along $C$. When $C$ is non-closed, in the authors’ previous works, the local existence of three distinct cuspidal edges along $C$ whose first fundamental forms coincide with that of $f$ was shown, under a certain reasonable assumption on $f$. In this paper, if $C$ is closed, that is, $C$ is a knot, we show that there exist infinitely many cuspidal edges along $C$ having the same first fundamental form as that of $f$ such that their images are non-congruent to each other, in general.

We generalize Yasuda’s examples of ribbon 2-knots of 1-fusion with different ribbon presentations.

We call a link (knot) $L$ to be strongly Jones (respectively, Homfly) undetectable, if there are infinitely many links which are not isotopic to $L$ but share the same Jones (respectively, Homfly) polynomial as $L$. We reconstruct Kanenobu’s knot [Kanenobu, Infinitely many knots with the same polynomial invariant, Proc. Amer. Math. Soc. 97(1) (1986), 158–162] and give two new constructions. Using these constructions, we give some examples of strongly Jones undetectable: $8_8$, $8_9$, $10_{22}$, $10_{35}$, $10_{155}$, $4_1\#4_1$, $5_2\#5_2!$ ($5_2!$ is the mirror image of $5_2$) and etc. For some special cases, these constructions will be shown to be strongly Jones undetectable and strongly Homfly undetectable.

In this paper, we describe the equivalence classes of simple arcs between the two punctures on a 2-punctured torus ${\mathrm{\Sigma }}_{1,2}$ up to isotopy by using the given four generators $g_1,g_2,g_3$ and $g_4$. Actually, we show that a class of simple arcs is represented by an ordered sequence of four integers. Also, we introduce an algorithm to check whether or not an ordered sequence of four integers represents a class of simple arcs in ${\mathrm{\Sigma }}_{1,2}.$ We want to point out that this result classifies the $(1,1)$-positions.

Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this paper, we discuss a categorified version of Vassiliev skein relation on Khovanov homology. More precisely, we will show that the “genus-one” operation gives rise to a crossing change on Khovanov complexes. Invariance under Reidemeister moves turns out, and it enables us to extend Khovanov homology to singular links. We then see that a long exact sequence of Khovanov homology groups categorifies Vassiliev skein relation for the Jones polynomials. In particular, the Jones polynomial is recovered even for singular links. We in addition discuss the FI relation on Khovanov homology.

For any alternating prime knot $K$, it is expected that the canonical genus of its Whitehead double is equal to the crossing number of $K$. We introduce local moves of alternating knots, and prove that these local moves preserve the canonical genus of its Whitehead double under a certain condition. By this result, we give a new family of knots which satisfy this conjecture.

Suciu constructed infinitely many ribbon 2-knots in $S^4$ whose knot groups are isomorphic to the trefoil knot group. They are distinguished by the second homotopy groups. We classify these knots by using $\mathrm{\text{SL}}(2,ℂ)$-representations of the fundamental groups of the 2-fold branched covering spaces.