Puiseux Expansion of a Cuspidal Singularity
We present an effective and elementary method of determining the topological type of a cuspidal plane curve singularity with given local parametrization.
Number of singular points of an annulus in
Using BMY inequality and a Milnor number bound we prove that any algebraic annulus in with no self-intersections can have at most three cuspidal singularities.
Hodge–type structures as link invariants
Based on some analogies with the Hodge theory of isolated hypersurface singularities, we define Hodge–type numerical invariants of any, not necessarily algebraic, link in a three–sphere. We call them
Geometry of Puiseux expansions
We consider the space Curv of complex affine lines t ↦ (x,y) = (ϕ(t),ψ(t)) with monic polynomials ϕ, ψ of fixed degrees and a map Expan from Curv to a complex affine space Puis with dim Curv = dim Puis, which is defined by initial Puiseux coefficients of the Puiseux expansion of the curve at infinity. We present some unexpected relations between geometrical properties of the curves (ϕ,ψ) and singularities of the map Expan. For example, the curve (ϕ,ψ) has a cuspidal singularity iff it is a critical...
On the Signatures of Torus Knots
We study properties of the signature function of the torus knot . First we provide a very elementary proof of the formula for the integral of the signature over the circle. We also obtain a closed formula for the Tristram-Levine signature of a torus knot in terms of Dedekind sums.
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