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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.
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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.
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.
The writhe polynomial is a fundamental invariant of an oriented virtual knot. We introduce a kind of local moves for oriented virtual knots called shell moves. The first aim of this paper is to prove that two oriented virtual knots have the same writhe polynomial if and only if they are related by a finite sequence of shell moves. The second aim of this paper is to classify oriented $2$-component virtual links up to shell moves by using several invariants of virtual links.
Multicrossings, which have previously been defined for classical knots and links, are extended to virtual knots and links. In particular, petal diagrams are shown to exist for all virtual knots.
We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid torus, thus answering a question posed by Arnold. We similarly study spaces of unframed links in the 3-sphere, modulo rotations, and spaces of knots in the thickened torus. The subgroup of meridional rotations splits as a direct factor of the fundamental group of the space of any framed link except the unknot. Its generators can be viewed as generalizations of the Gramain loop in the space of long knots. Taking the quotient by certain such rotations relates the spaces we study. All of our results generalize previous work of Hatcher and Budney. We provide many examples and explicitly describe generators of fundamental groups.
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