We go along a knot diagram, and get a sequence of over-and under-crossing points. We will study which kinds of sequences are realized by diagrams of the trefoil knot. As an application, we will characterize the Shimizu warping polynomials for diagrams of the trefoil knot.
We introduce three kinds of invariants of a virtual knot called the first, second, and third intersection polynomials. The definition is based on the intersection number of a pair of curves on a closed surface. We study properties of the intersection polynomials and their applications concerning the behavior on symmetry, the crossing number and the virtual crossing number, a connected sum of virtual knots, characterizations of intersection polynomials, finite type invariants of order two, and a flat virtual knot.
For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an descending diagram. We study relations between the ascending number and geometrical invariants; the crossing number, the genus and the bridge index. The main aim of this paper is to show that there exists a knot [Formula: see text] such that [Formula: see text] and [Formula: see text], and that there exists a knot [Formula: see text] such that [Formula: see text] and [Formula: see text] for any positive integer [Formula: see text]. We also give an upper bound of the ascending number for a [Formula: see text]-bridge knot.
For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an ascending diagram. We study the ascending number of a knot by analyzing the Conway polynomial. In this paper, we give a sharper lower bound of the ascending number of a knot and newly determine the ascending number for 26 prime knots up to 10 crossings.
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