Journal of Knot Theory and Its Ramifications

Knot Floer homology is an invariant for knots in the three-sphere for which the Euler characteristic is the Alexander–Conway polynomial of the knot. The aim of this paper is to study this homology for a class of satellite knots, so as to see how a certain relation between the Alexander–Conway polynomials of the satellite, companion and pattern is generalized on the level of the knot Floer homology. We also use our observations to study a classical geometric invariant, the Seifert genus, of our satellite knots.

We discuss the relation between arc index, maximal Thurston–Bennequin number, and Khovanov homology for knots. As a consequence, we calculate the arc index and maximal Thurston–Bennequin number for all knots with at most 11 crossings. For some of these knots, the calculation requires a consideration of cables which also allows us to compute the maximal self-linking number for all knots with at most 11 crossings.

The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This defines a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister moves to give an invariant volume on the Khovanov homology. In this paper, its construction and invariance under these moves is demonstrated. Also, some examples of the invariant are presented for particular choices for the bases of homology groups to obtain a numerical invariant of knots and links. In these examples, the algebraic torsion seen in the Khovanov chain complex when homology is computed over ℤ is recovered.

Let G be a 0- or 1-deficient graph which is intrinsically knotted, let J represent any graph obtained from G by a finite sequence of Δ-Y exchanges and/or vertex expansions. We prove that removing any vertex of J, and all edges incident to that vertex, yields an intrinsically linked graph. This result provides more intrinsically knotted graphs which satisfy the conjecture mentioned in Adams' The Knot Book that removing any vertex from an intrinsically knotted graph yields an intrinsically linked graph.

A handlebody-knot is a handlebody embedded in the 3-sphere. We enumerate all genus two handlebody-knots up to six crossings.

An oriented and ordered n-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring orientations and ordering of the components. For an oriented and ordered n-component link L, let λ_L be the product of linking numbers of all 2-component sublinks in L. For n = 4m + 3, where m is a non-negative integer, we show that if L is achiral then λ_L = 0. And for n ≠ 4m + 3, we show that there exists an n-component achiral link L with λ_L ≠ 0 by using achiral embeddings of complete graphs with n vertices K_n.

In this paper, we give a relationship among the period, the minimal supporting genus vg(L) of a periodic virtual link L that admits a minimal supporting genus periodic diagram, and vg(L_*) of its factor link L_*. As an application, it is shown that if L is a non-classical virtual link of period n which admits a minimal supporting genus n-periodic diagram, then n ≤ 2vg(L) + 1, except a certain class of periodic virtual links.

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point p of the cylinder is called coherent if all three branches intersect at p pairwise with the same intersection index. A triple unknotting of a classical knot K is a homotopy which connects K with the trivial knot and which has as singularities only coherent triple points.

We give a new formula for the first Vassiliev invariant v₂(K) by using triple unknottings. As a corollary we obtain a very simple proof of the fact that passing a coherent triple point always changes the knot type. As another corollary we show that there are triple unknottings which are not homotopic as triple unknottings even if we allow more complicated singularities to appear in the homotopy of the triple homotopy.

In a prequel to this article, we used abstract Wiener measure to define the Chern–Simons path integral over ℝ³. In this sequel, we compute the Wilson Loop observable for the non-abelian gauge group and compare with current knot literature.

Let S be a bordered orientable Klein surface and p a prime. Assume that f is an order p automorphism of S. In this work we obtain the conditions on the topological type of (S, f) to be conformally equivalent to (S′, f′) where S′ is a bordered orientable Klein surface embedded in the Euclidean space and f′ is the restriction to S′ of a prime order rotation. Our results can allow the visualization of some Riemann surfaces automorphisms by representing them as restrictions of isometries of 𝕊⁴ or ℝ⁴. We illustrate this method with the order seven automorphisms of two famous Riemann surfaces: the Klein quartic and the Wiman surface.

In this paper, colorings by symmetric quandles for spatial graphs and handlebody-links are introduced. We also introduce colorings by LH-quandles for LH-links. LH-links are handlebody-links, some of whose circle components are specified, which are related to Heegaard splittings of link exteriors. We also discuss quandle (co)homology groups and cocycle invariants.

Gusarov and Habiro introduced a C_m move, that is strongly related to Vassiliev invariants. In this note, we study a special kind of C_m move, called a non-self C_m move. We show that two links can be transformed into each other by a finite sequence of non-self C_m moves if and only if (1) the two links can be transformed into each other by a finite sequence of C_m moves, and (2) the knot types of corresponding components coincide.