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Register a new userColoring invariants of knots and links are often intractable
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Added by
Greg Kuperberg
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Authors
Greg Kuperberg
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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.
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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.
It is shown that every knot or link is the set of complex tangents of a 3-sphere smoothly embedded in the three-dimensional complex space. We show in fact that a one-dimensional submanifold of a closed orientable 3-manifold can be realised as the set of complex tangents of a smooth embedding of the 3-manifold into the three-dimensional complex space if and only if it represents the trivial integral homology class in the 3-manifold. The proof involves a new application of singularity theory of differentiable maps.
Quandle coloring quivers are directed graph-valued invariants of oriented knots and links, defined using a choice of finite quandle $X$ and set $Ssubsetmathrm{Hom}(X,X)$ of endomorphisms. From a quandle coloring quiver, a polynomial knot invariant known as the textit{in-degree quiver polynomial} is defined. We consider quandle coloring quiver invariants for oriented surface-links, represented by marked graph diagrams. We provide example computations for all oriented surface-links with ch-index up to 10 for choices of quandles and endomorphisms.
We prove the equivalence of the invariants EH(L) and LOSS-(L) for oriented Legendrian knots L in the 3-sphere equipped with the standard contact structure, partially extending a previous result by Stipsicz and Vertesi. In the course of the proof we relate the sutured Floer homology groups associated with a knot complement and knot Floer homology, and define intermediate Legendrian invariants.
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.
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