1
vote
0answers
214 views

subset embedding gives trefoil knot

Let $X$ be a topological space and $E_n(X)$ the space of finite sets of cardinality $\leq n$. It is a theorem of Bott that $E_3(S^1)=S^3$. What is the idea to show that the embedding ...
3
votes
0answers
334 views

Positivity of braid monodromy of curve singularities

I recall the notion of braid monodromy. Let $C \subset \mathbb{C}^2$ be an algebraic curve, and choose a projection $\pi: \mathbb{C}^2 \to \mathbb{C}$ such that the restriction $\pi: C \to ...
5
votes
1answer
540 views

Link of singularities

For an isolated plane curve singularity, given by homogeneous equation $F=0 \subset \mathbb{C}^2$, one consider the curve $(F=0) \cap S^3 \subset S^3$, and we call it the link of singularity. some ...
4
votes
1answer
859 views

Genera and the Milnor Conjecture on the Unknotting Number of a Torus Knot

Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = ...
13
votes
4answers
823 views

What are the points of Spec(Vassiliev Invariants)?

Background Recall that a (oriented) knot is a smoothly embedded circle $S^1$ in $\mathbb R^3$, up to some natural equivalence relation (which is not quite trivial to write down). The collection of ...