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Register a new userPolynomial Invariants of Singular Knots and links
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We generalize the notion of the quandle polynomial to the case of singquandles. We show that the singquandle polynomial is an invariant of finite singquandles. We also construct a singular link invariant from the singquandle polynomial and show that this new singular link invariant generalizes the singquandle counting invariant. In particular, using the new polynomial invariant, we can distinguish singular links with the same singquandle counting invariant.
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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 extend the quandle cocycle invariant to oriented singular knots and links using algebraic structures called emph{oriented singquandles} and assigning weight functions at both regular and singular crossings. This invariant coincides with the classical cocycle invariant for classical knots but provides extra information about singular knots and links. The new invariant distinguishes the singular granny knot from the singular square knot.
Kashaev and Reshetikhin previously described a way to define holonomy invariants of knots using quantum $mathfrak{sl}_2$ at a root of unity. These are generalized quantum invariants depend both on a knot $K$ and a representation of the fundamental group of its complement into $mathrm{SL}_2(mathbb{C})$; equivalently, we can think of $mathrm{KR}(K)$ as associating to each knot a function on (a slight generalization of) its character variety. In this paper we clarify some details of their construction. In particular, we show that for $K$ a hyperbolic knot $mathrm{KaRe}(K)$ can be viewed as a function on the geometric component of the $A$-polynomial curve of $K$. We compute some examples at a third root of unity.
Let $G$ be a nonabelian, simple group with a nontrivial conjugacy class $C subseteq G$. Let $K$ be a diagram of an oriented knot in $S^3$, thought of as computational input. We show that for each such $G$ and $C$, the problem of counting homomorphisms $pi_1(S^3setminus K) to G$ that send meridians of $K$ to $C$ is almost parsimoniously $mathsf{#P}$-complete. This work is a sequel to a previous result by the authors that counting homomorphisms from fundamental groups of integer homology 3-spheres to $G$ is almost parsimoniously $mathsf{#P}$-complete. Where we previously used mapping class groups actions on closed, unmarked surfaces, we now use braid group actions.
We introduce shadow structures for singular knot theory. Precisely, we define emph{two} invariants of singular knots and links. First, we introduce a notion of action of a singquandle on a set to define a shadow counting invariant of singular links which generalize the classical shadow colorings of knots by quandles. We then define a shadow polynomial invariant for shadow structures. Lastly, we enhance the shadow counting invariant by combining both the shadow counting invariant and the shadow polynomial invariant. Explicit examples of computations are given.
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