Ayaka Shimizu

THE WARPING DEGREE OF A KNOT DIAGRAM

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).

THE HALF-TWISTED SPLICE OPERATION ON REDUCED KNOT PROJECTIONS

We show that any nontrivial reduced knot projection can be obtained from a trefoil projection by a finite sequence of half-twisted splice operations and their inverses such that the result of each step in the sequence is reduced.

THE WARPING POLYNOMIAL OF A KNOT DIAGRAM

We introduce the warping polynomial of an oriented knot diagram. In this paper, we characterize the warping polynomial, and define the span of a knot to be the minimal span of the warping polynomial for all diagrams of the knot. We show that the span of a knot is one if and only if it is non-trivial and alternating, and we give an inequality between the span and the dealternating number.

The warping degree of a link diagram

For an oriented link diagram D, the warping degree d(D) is the smallest number of crossing changes which are needed to obtain a monotone diagram from D. We show that d(D)+d(-D)+sr(D) is less than or equal to the crossing number of D, where -D denotes the inverse of D and sr(D) denotes the number of components which have at least one self-crossing. Moreover, we give a necessary and sufficient condition for the equality. We also consider the minimal d(D)+d(-D)+sr(D) for all diagrams D. For the warping degree and linking warping degree, we show some relations to the linking number, unknotting number, and the splitting number.

Quantization of the Crossing Number of a Knot Diagram

We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for all the states for the plane curve. As an application, we show that every closed transversely intersected plane curve with even crossing points has two independent canonical orientations and every based closed transversely intersected plane curve with odd crossing points has two independent canonical orientations.

Up–down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type II

We introduce an up–down coloring of a virtual-link (or classical-link) diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two 2-component virtual-link (or classical-link) diagrams. By using the notion of a quandle cocycle invariant, we give a method to detect the necessity of Reidemeister moves of type II between two given virtual-knot (or classical-knot) diagrams. As an application, we show that for any virtual-knot diagram D, there exists a diagram D′ representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister move of type II.

On the orientations of monotone knot diagrams

An oriented monotone knot diagram is a knot diagram such that one meets each crossing as an over-crossing first as one travels the diagram with the orientation by starting at a point on the diagram. In this paper, unoriented knot projections which are monotone with an orientation and any over/under information are characterized. Also, monotone diagrams which are monotone with exactly one orientation and unique basepoint are characterized. As an application, a necessary condition for a knot projection with reductivity four is given.

Region crossing change on spatial-graph diagrams

Region crossing change at a region of a knot, link or spatial-graph diagram is a local transformation which changes all the crossings on the boundary of the region. In this paper, we show that we can make any crossing change by a finite number of region crossing changes on any diagram of a connected spatial graph which has no cutting circles.

Axis systems of link projections

An axis of a link projection is a closed curve which lies symmetrically on each region of the link projection. In this paper we define axis systems of link projections and characterize axis systems of the standard projections of twist knots.