---
tag: C831AN3RFM
kind: concept
name: Link of a singularity
msc: [32S55, 32S50, 14B05]
sources:
  - url: https://en.wikipedia.org/wiki/Milnor_map
---

## Summary

The link of a singular point of a complex analytic set is its intersection with a small sphere centred at the point, see the [[definition|MHGZEW6RBM]].
Near the point, the set is homeomorphic to the cone over its link, so the link carries the local topology of the singularity.
For an [[isolated hypersurface singularity|VTGYEES3VS]] in $\C^{n+1}$ the link is a smooth closed manifold of dimension $2n - 1$.
For plane curves it is a [[link|1A5WRCNTM3]] in the knot theory sense, a disjoint union of circles in $S^3$.

## Explanations

### L1 human

To see what a singular point looks like, cut the space with a tiny sphere around it and look at what is left on the sphere.
The space near the point is then the cone over this cut, made of rays from the point to the cut.
For the curve $x^2 = y^3$ in $\C^2$, the cut is a knotted circle in the $3$-sphere, the trefoil knot.
For two lines crossing at a point, it is two circles that link once, the Hopf link.