Narrow Results

We introduce a family of polynomial invariants for knotoids. Via the corresponding Gauss diagrams for knotoids, we investigate the method to calculate these polynomial invariants. Also, we give some properties and examples.

This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relation to these phenomena. Information on determinations of the four-ball genus of some virtual knots is obtained by via the affine index polynomial in conjunction with results on the genus of positive virtual knots using joint work with Dye and Kaestner.

A multi-variable polynomial invariant for knotoids and linkoids, which is an enhancement of the bracket polynomial for knotoids introduced by Turaev, is given by using the concept of a pole diagram which originates in constructing a virtual link invariant. Several features of the polynomial are revealed.

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.

The height of a knotoid is a measure of how far it is from being a knot. Here, we define the positive and negative parts of the height, and we prove that they determine the unsigned height. Some polynomial invariants provide lower bounds for the signed heights. We also study a set of sequences associated to a knotoid.

Consider two parallel lines ${\ell }_1$ and ${\ell }_2$ in ${ℝ}^3$. A rail arc is an embedding of an arc in ${ℝ}^3$ such that one endpoint is on ${\ell }_1$, the other is on ${\ell }_2$, and its interior is disjoint from ${\ell }_1\cup {\ell }_2$. Rail arcs are considered up to rail isotopies, ambient isotopies of ${ℝ}^3$ with each self-homeomorphism mapping ${\ell }_1$ and ${\ell }_2$ onto themselves. When the manifolds and maps are taken in the piecewise linear category, these rail arcs are called stick rail arcs. The stick number of a rail arc class is the minimum number of sticks, line segments in a p.l. arc, needed to create a representative. This paper calculates the stick number of rail arcs classes with a crossing number at most 2 and uses a winding number invariant to calculate the stick numbers of infinitely many rail arc classes. Each rail arc class has two canonically associated knot classes, its under and over companions. This paper also introduces the rail stick number of knot classes, the minimum number of sticks needed to create a rail arcs whose under or over companion is the knot class. The rail stick number is calculated for 29 knot classes with crossing number at most 9. The stick number of multi-component rail arcs classes is considered as well as the lattice stick number of rail arcs.

In this paper, we construct a polynomial invariant of Kauffman type for kd($1$)-linkoids and compute the polynomials for knotoids with up to three crossings. As a consequence, it is shown that the polynomial is different from the previous one in [Y. Miyazawa, A polynomial invariant of Kauffman type for knotoids, preprint].

As a generalization of the classical knots, knotoids deal with the open ended knot diagram in a surface. In recent years, many polynomial invariants for knotoids appeared, such as the bracket polynomial, the index polynomial and the $n$th polynomial. In this paper, we introduce a new polynomial invariant $F$-polynomial for knotoids and discuss some properties of the $F$-polynomial. Finally, we construct a family of knotoid diagrams which can be distinguished from each other by the $F$-polynomial but cannot be distinguished by the index polynomial and the $n$th polynomial.