Journal of Knot Theory and Its Ramifications

For an oriented knot diagram D, the warping degree d(D) is the smallest number of crossing changes which are needed to obtain the monotone diagram from D in the usual way. We show that d(D) + d(-D) + 1 is less than or equal to the crossing number of D. Moreover, the equality holds if and only if D is an alternating diagram. For a knot K, we also estimate the minimum of d(D) + d(-D) for all diagrams D of K with c(D) = c(K).

We extend the Kauffman state models of the Jones and Alexander polynomials of classical links to state models of their two-variable extensions in the case of singular links. Moreover, we extend both of them to polynomials with d + 1 variables for long singular knots with exactly d double points. These extensions can detect non-invertibility of long singular knots.

It is well known that any virtual link is described as the closure of a virtual braid. Therefore, we can define the virtual braid index. Ohyama proved an inequality for the crossing number and the braid index of a classical link. In this paper, we prove an analogous inequality for the (total) crossing number and the braid index of a virtual link.

Let K be a two-bridge knot or link in S³. Then K is also denoted as the four-plat, b(p, q) to indicate its association with some rational number p/q. The lens space L = L(p, q) admits an isometry τ of order two, such that the quotient space L modulo the involution τ is an orbifold whose exceptional set is K and whose underlying space is S³. In this paper, the mapping class groups of these orbifolds are classified. While these groups can be found as a result of Mostow's Rigidity Theorem, this paper calculates the generators and relations of the groups and the proof does not rely on this strong theorem for the majority of cases.

It is well-known that for any link L, twice the unknotting number of L is less than or equal to the crossing number of L. Taniyama characterized the links which satisfy the equality. We characterize the links where twice the unknotting number is equal to the crossing number minus one. As a corollary, we show that for any link L with twice the unknotting number of L is greater than or equal to the crossing number of L minus two, every minimal diagram of L realizes the unknotting number.

This note establishes a stronger version of a conjecture of Reid and others in the arithmetic case: if two elements of equal trace (e.g. conjugate elements) generate an arithmetic two-bridge knot or link group, then the elements are parabolic (and hence peripheral). This includes the figure-eight knot and Whitehead link groups. Similarly, if two conjugate elements generate the trefoil knot group, then the elements are peripheral.

A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper, we show that if a generic immersion of a planar graph is knotted then the number of double points of the immersion is more than or equal to three. To prove this, we also show that an embedding of a graph obtained from a generic immersion of the graph (does not need to be planar) with at most three double points is totally free if it contains neither a Hopf link nor a trefoil knot.

Two welded (respectively virtual) link diagrams are homotopic if one may be transformed into the other by a sequence of extended Reidemeister moves, classical Reidemeister moves, and self crossing changes. In this paper, we extend Milnor's μ and invariants to welded and virtual links. We conclude this paper with several examples, and compute the μ invariants using the Magnus expansion and Polyak's skein relation for the μ invariants.

In this paper, we prove that an adequate virtual link diagram of an adequate virtual link has minimal real crossing number.