A relation between the crossing number and the height of a knotoid

Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such arcs which are disjoint from crossings. In the paper, we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knotoid is greater than or equal to twice the height of the knotoid. Combining the inequality with known lower bounds of the height we obtain a lower bounds of the crossing number of a knotoid via the extended bracket polynomial, the affine index polynomial and the arrow polynomial of the knotoid. As an application of our result we prove an upper bound for the length of a bridge in a minimal diagram of a classical knot: the number of crossings in a minimal diagram of a knot is greater than or equal to three times the length of a longest bridge in the diagram.

On the warping sum of knots

The warping sum [Formula: see text] of a knot [Formula: see text] is the minimal value of the sum of the warping degrees of a minimal diagram of [Formula: see text] with both orientations. In this paper, knots [Formula: see text] with [Formula: see text] are characterized, and some knots [Formula: see text] with [Formula: see text] are given.

ON AN INEQUALITY BETWEEN UNKNOTTING NUMBER AND CROSSING NUMBER OF LINKS

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.

A PARTIAL ORDERING OF KNOTS AND LINKS THROUGH DIAGRAMMATIC UNKNOTTING

In this paper we define a partial ordering of knots and links using a special property derived from their minimal diagrams. A link [Formula: see text] is called a predecessor of a link [Formula: see text] if [Formula: see text] and a diagram of [Formula: see text] can be obtained from a minimal diagram D of [Formula: see text] by a single crossing change. In such a case, we say that [Formula: see text]. We investigate the sets of links that can be obtained by single crossing changes over all minimal diagrams of a given link. We show that these sets are specific for different links and permit partial ordering of all links. Some interesting results are presented and many questions are raised.

On Kauffman Brackets

An antichain is associated to each link diagram so that the highest degree of the Kauffman bracket can be determined. As an application, we show that the span of the Kauffman bracket is less than or equal to 4(n - m) for dealternator connected m-alternating diagrams and the upper bound is best possible. This completely solves a conjecture of [1]. Finally, we show that a semi-alternating diagram may be not a minimal diagram which disproves a conjecture of K. Murasugi [10].