Singular Knots
Singular Knot
A singular knot is a smooth map
which fails to be an embedding, where the only failures allowed are simple double points. That is, places where the curve intersects itself transversally, and at the intersection only two threads intersect.
Attention
We will restrict our attention to oriented singular knots, which are equipped with a direction.
Singular Knot Diagram
A singular knot diagram is a planar projection of a singular knot, which has two types of crossings: regular crossings with over/under strand information, and double points.
Each singularity of a singular knot can be resolved two ways: by replacing the double point with an over-crossing, or with an under-crossing. So we can define a resolution map as follows:
Resolution Map
Let
denote the set of -singular knots. The resolution map is defined by replacing each singularity of a given -singular knot by the difference of its two resolutions.
The Vassiliev Filtration
Vassiliev Filtration
Suppose
is the set of all oriented knots , consider the algebra of formal linear combinations. The Vassiliev-Goussarour filtration is a filtration of by the number of double points in a singular knot diagram resulting from the inverse resolution map: where is the ideal spanned by knots with at least double points.
e.g. We can compute the trefoil minus unknot as follows:
Remark
is an algebra by linearly extending the commutative monoid structure. i.e. make connected sum distributive.