Narrow Results

We bring cocycle enhancement theory to the case of psyquandles. Analogously to our previous work on virtual biquandle cocycle enhancements, we define enhancements of the psyquandle counting invariant via pairs of a biquandle 2-cocycle and a new function satisfying some conditions. As an application we define new single-variable and two-variable polynomial invariants of oriented pseudoknots and singular knots and links. We provide examples to show that the new invariants are proper enhancements of the counting invariant and are not determined by the Jablan polynomial.

In this paper we extend the idea of Dehn surgery to a singular knot with one singularity, and give conditions under which the surgery manifold of an untangled singular knot is a Haken manifold containing an incompressible torus. We then show that untangled singular knots have Property P.

In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Y_d,n(u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SB_n into the algebra Y_d,n(u). Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra Y_d,n(u).

In this paper, we introduce pseudo braid monoids and show that, such a monoid is isomorphic to a singular braid monoid. We also prove an analogue of Markov’s Theorem for pseudo braids.

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.

Given a biquandle $(X,S)$, a function $\tau$ with certain compatibility and a pair of non commutative cocyles$f,h:X\times X\to G$ with values in a non necessarily commutative group $G$, we give an invariant for singular knots/links. Given $(X,S,\tau )$, we also define a universal group $U_{nc}^{fh}(X)$ and universal functions governing all 2-cocycles in $X$, and exhibit examples of computations. When the target group is abelian, a notion of abelian cocycle pair is given and the “state sum” is defined for singular knots/links. Computations generalizing linking number for singular knots are given. As for virtual knots, a “self-linking number” may be defined for singular knots.

In this paper, we deal with the notion of singquandles introduced in [I. R. U. Churchill, M. Elhamdadi, M. Hajij and S. Nelson, Singular knots and involutive quandles, J. Knot Theory Ramifications 26(14) (2017) 1750099]. This is an algebraic structure that naturally axiomatizes Reidemeister moves for singular links, similarly to what happens for ordinary links and quandles. We present a new axiomatization that shows different algebraic aspects and simplifies applications. We also reformulate and simplify the axioms for affine singquandles (in particular in the idempotent case).

We introduce algebraic structures known as psybrackets and use them to define invariants of pseudoknots and singular knots and links. Psybrackets are Niebrzydowski tribrackets with additional structure inspired by the Reidemeister moves for pseudoknots and singular knots. Examples and computations are provided.

We enhance the psyquandle counting invariant for singular knots and pseudoknots using quivers analogously to quandle coloring quivers. This enables us to extend the in-degree polynomial invariants from quandle coloring quiver theory to the case of singular knots and pseudoknots. As a side effect, we obtain biquandle coloring quivers and in-degree polynomial invariants for classical and virtual knots and links.

Starting from a Weight-System denoted by P and defined on the n-Chord-Diagrams with values in an arbitrary Q–module, we give an explicit combinatorial formula for an invariant of (n–1)-singular knots which has P as its derivative. The formula is defined for regular knot projections. Its invariance under singular Reidemeister moves is then proved.