Journal of Knot Theory and Its Ramifications

Recently Kearton showed that any Seifert matrix of a knot is S-equivalent to the Seifert matrix of a prime knot. We show in this note that such a matrix is in fact S-equivalent to the Seifert matrix of a hyperbolic knot. This result follows from reinterpreting this problem in terms of Blanchfield pairings and by applying results of Kawauchi.

A finite set of specific knotted double-handcuff graphs is shown to be minimal among those which produce all projections of knotted double-handcuff graphs. In addition, we show that a double-handcuff graph has no strongly almost trivial spatial embeddings.

In this paper we give a lower bound of triple point numbers of special family of 2-knots colored by the dihedral quandle of order 5.

A ribbon 2-knot is a 2-sphere in R⁴, which is obtained from m 2-spheres in R⁴ by connecting them with m - 1 pipes. Even in the case that a ribbon 2-knot K² is constructed by two 2-spheres and one pipe, a way of constructing K² is not always unique. That is, K² might have presentations of two or more types.

The presentation that a pipe crosses two spheres n times is said to be an n-crossing ribbon presentation, and also n be the ribbon crossing number of this presentation.

In this note, we will show that for any positive integer i, there exists a ribbon 2-knot with presentations of two distinct types such that the difference of their ribbon crossing number is i; and in addition we will also show that for any positive integer j, there exists a ribbon 2-knot with presentations of j, distinct types such that they have the same ribbon crossing number.

The thickness, NIR(K) of a knot or link K is defined to be the radius of the largest open solid tube one can put around the curve without any self intersections of the normal discs, which is also known as the normal injectivity radius of K. For C^1,1 curves K, , where κ(K) is the generalized curvature, and the double critical self distance DCSD(K) is the shortest length of the segments perpendicular to K at both end points. The knots and links in ideal shapes (or tight knots or links) belong to the minima of ropelength = length/thickness within a fixed isotopy class. In this article, we prove that NIR(K) = ½DCSC(K), for every relative minimum K of ropelength in Rⁿ for certain dimensions n, including n = 3.

A free self delta-trivial link is a link which can be transformed into a trivial link by trivializing a component only with self delta-moves on the component, one by one in any order. We show that if a link spans mutually disjoint disks whose singularities are arcs of the non-clasp-type and loops, then the link is free self delta-trivial. We also study free self sharp-triviality and free self pass-triviality of a link which spans mutually disjoint disks whose singularities are mutually disjoint simple arcs of the clasp-type.

In this paper, we introduce a representation of knots and links in S³ by integral matrices and then give an explicit formula for the Casson invariant for integral homology 3-spheres obtained from S³ by Dehn surgery along the knots and links represented by the integral matrices in which either all entries are even or the entries of each row are the same odd number. As applications, we study the preimage of the Casson invariant for a given integer and also give formulas for the Casson invariants of some special classes of integral homology 3-spheres.

We define a local move called a CF-move on virtual link diagrams, and show that any virtual knot can be deformed into a trivial knot by using generalized Reidemeister moves and CF-moves. Moreover, we define a new virtual link invariant n(L) for a virtual 2-component link L whose virtual linking number is an integer. Then we give necessary and sufficient conditions for two virtual 2-component links to be deformed into each other by using generalized Reidemeister moves and CF-moves in terms of a virtual linking number and n(L).