Tangle equations involving Montesinos links

Generalized Montesinos tangles are classified, and the system of unoriented tangle equations [Formula: see text] and [Formula: see text] is solved for a generalized Montesinos tangle [Formula: see text] where [Formula: see text] and [Formula: see text] are rational tangles and [Formula: see text] and [Formula: see text] are Montesinos knots/links.

Labeled singular knots

We introduce labeled singular knots and equivalently labeled 4-valent rigid vertex spatial graphs. Labeled singular knots are singular knots with labeled singularities. These knots are considered subject to isotopies preserving the labelings. We provide a topological invariant schema similar to that of Henrich and Kauffman in [A. Henrich and L. H. Kauffman, Tangle insertion invariants for pseudoknots, singular knots, and rigid vertex spatial graphs, Contemp. Math. 689 (2017) 1–10] by inserting rational tangles at the labeled singularities to extend usual knot invariants to our class of singular knots. We show that we can use invariants of labeled singular knots to serve usual singular knots. Labeled framed pseudoknots are also introduced and discussed.

Virtual rational tangles

We use Kauffman’s bracket polynomial to define a complex-valued invariant of virtual rational tangles that generalizes the well-known fraction invariant for classical rational tangles. We provide a recursive formula for computing the invariant, and use it to compute several examples.

A bridge between Conway–Coxeter friezes and rational tangles through the Kauffman bracket polynomials

In this paper, we build a bridge between Conway–Coxeter friezes (CCFs) and rational tangles through the Kauffman bracket polynomials. One can compute a Kauffman bracket polynomial attached to rational links by using CCFs. As an application, one can give a complete invariant on CCFs of zigzag-type.

Algebraic and topological structures on rational tangles

<p>In this paper we present the construction of a group Hopf algebra on the class of rational tangles. A locally finite partial order on this class is introduced and a topology is generated. An interval coalgebra structure associated with the locally finite partial order is specified. Irrational and real tangles are introduced and their relation with rational tangles are studied. The existence of the maximal real tangle is described in detail.</p>

Foliated compressing discs for Legendrian rational tangles

We establish a new framework for diagramming both Legendrian rational tangles in the standard contact structure on [Formula: see text] and the signed characteristic foliations of their associated compressing discs, as well as the technical means by which these diagrams can be used to study Legendrian isotopies of such tangles. We then establish a number of results that represent new progress in the ongoing effort to classify Legendrian rational tangles under a pair of operations known as Legendrian flypes. These operations, while topologically isotopies, are known to produce distinct Legendrian objects in many circumstances, a fact that has been of much interest throughout the study and classification of Legendrian knots.

A note on quasi-alternating Montesinos links

Quasi-alternating links are a generalization of alternating links. They are homologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the classification of quasi-alternating pretzel links in terms of their integer tassel parameters. Replacing tassels by rational tangles generalizes pretzel links to Montesinos links. In this paper we establish conditions on the rational parameters of a Montesinos link to be quasi-alternating. Using recent results on left-orderable groups and Heegaard Floer L-spaces, we also establish conditions on the rational parameters of a Montesinos link to be non-quasi-alternating. We discuss examples which are not covered by the above results.