Narrow Results

In this paper, we consider biquandle colorings for knotoids in ${ℝ}^2$ or $S^2$, and we construct several coloring invariants for knotoids derived as enhancements of the biquandle counting invariant. We first enhance the biquandle counting invariant by using a matrix constructed by utilizing the orientation a knotoid diagram is endowed with. We generalize Niebrzydowski’s biquandle longitude invariant for virtual long knots to obtain new invariants for knotoids. We show that biquandle invariants can detect mirror images of knotoids and show that our enhancements are proper in the sense that knotoids which are not distinguished by the counting invariant are distinguished by our enhancements.

Biquandle brackets are a type of quantum enhancement of the biquandle counting invariant for oriented knots and links, defined by a set of skein relations with coefficients which are functions of biquandle colors at a crossing. In this paper, we use biquandle brackets to enhance the biquandle counting matrix invariant defined by the first two authors in (N. Gügümcü and S. Nelson, Biquandle coloring invariants of knotoids, J. Knot Theory Ramif.28(4) (2019) 1950029). We provide examples to illustrate the method of calculation and to show that the new invariants are stronger than the previous ones. As an application we show that the trace of the biquandle bracket matrix is an invariant of the virtual closure of a knotoid.

In this paper, we study parity in planar and spherical knotoids in relation to virtual knots. We introduce a planar version of the parity bracket polynomial for planar knotoids. We show that the virtual closure map (a map from the set of knotoids in $S^2$ to the set of virtual knots of genus at most one) is not surjective, by utilizing the surface bracket polynomial of virtual knots. We give specific examples of virtual knots that are not in the image of the virtual closure map. Turaev conjectured that minimal diagrams of knot-type knotoids have zero height. We prove this conjecture by using the results of Nikonov and Manturov induced by parities of virtual knots.

In this paper, we introduce a new polynomial invariant of planar (but not spherical) knotoids, which we call the winding signed sum polynomial. This Laurent polynomial invariant of planar knotoids denoted by ${\beta }_K$ is a type-one Vassiliev invariant. This invariant might tell whether a planar knotoid is a zero-height or nonzero-height planar knotoid. It also might distinguish between a planar knotoid and its inverse, while the affine index polynomial defined by Gügümcü and Kauffman in [New invariants of knotoids, Eur. J. Combin.65 (2017) 186–229] (that is also a type-one Vassiliev invariant) cannot distinguish between a planar knotoid and its inverse. We also define some geometric invariants of planar knotoids, and give lower bounds for these invariants using the winding signed sum polynomial, which helps in computing these geometric invariants that are easy to define, but hard to calculate.

The entanglement of open curves in 3-space appears in many physical systems and affects their material properties and function. A new framework in knot theory was introduced recently, that enables to characterize the complexity of collections of open curves in 3-space using the theory of knotoids and linkoids, which are equivalence classes of diagrams with open arcs. In this paper, new invariants of linkoids are introduced via a surjective map between linkoids and virtual knots. This leads to a new collection of strong invariants of linkoids that are independent of any given virtual closure. This gives rise to a collection of novel measures of entanglement of open curves in 3-space, which are continuous functions of the curve coordinates and tend to their corresponding classical invariants when the endpoints of the curves tend to coincide.