Narrow Results

We introduce a new approach to Vassiliev invariants. This approach deals with Vassiliev invariants directly on knots and does not make use of diagrams. We give a series of applications of this approach, (re)proving some new and known facts on Vassiliev invariants.

In this paper, we classify, up to rigid isotopy, real rational knots of degrees less than or equal to $6$ in a real quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots of low degree in ${ℝ\mathbb{P}}^3$ and classify real rational curves of degree 6 in the 3-sphere with exactly one ordinary double point.

We give examples of a linear combination of algebraic knots and their mirrors that are algebraically slice, but whose topological and smooth four-genus is two. Our examples generalize an example of non-slice algebraically slice linear combination of iterated torus knots obtained by Hedden, Kirk and Livingston. Our main tool is a genus bound from Casson–Gordon theory and a cabling formula that allows us to compute effectively these invariants.