A virtual link may be defined as an equivalence class of diagrams, or alternatively as a
stable equivalence class of links in thickened surfaces. We prove that a minimal crossing …

CROSSING CHANGES, DELTA MOVES AND SHARP MOVES ON WELDED KNOTS 1.
Introduction. Virtual knots [9] and welded knots [3] are two ex Page 1 ROCKY MOUNTAIN …

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 …

The Only Undergraduate Textbook to Teach Both Classical and Virtual Knot Theory An
Invitation to Knot Theory: Virtual and Classical gives advanced undergraduate students a …

We show that for any nontrivial knot K and any natural number n, there is a diagram D of K
such that the unknotting number of D is greater than or equal to n. It is well-known that twice …

A knitting pattern is a tangle diagram in the square and a knitting is a tessellation of a knitting
pattern in the plane. In this paper, a classification of the knitting patterns is done in terms of …

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 …

This article is a revised detailed version of the research announcement [Bussei Kenkyu 92
(2009), 16] introducing a complexity of a spatial graph, which is useful to transform a spatial …

Let $\alpha $ be a map from the set of all knot types ${\mathcal K} $ to a set $ X $. Let $\beta
$ be a map from ${\mathcal K} $ to a set $ Y $. We define the relation between $\alpha $ and …

By considering unknotting operations, we obtain ways of measuring how knotted a knot is.
Unknotting phenomena can be seen not only in knot theory, but also in various settings such …