On knots in subdivergent diagrams

Sutured Heegaard diagrams for knots

We define sutured Heegaard diagrams for null-homologous knots in 3–manifolds. These diagrams are useful for computing the knot Floer homology at the top filtration level. As an application, we give a formula for the knot Floer homology of a Murasugi sum. Our result echoes Gabai’s earlier works. We also show that for socalled “semifibred" satellite knots, the top filtration term of the knot Floe...

A Partial Ordering of Knots Through Diagrammatic Unknotting

In this paper we define a partial order on the set of all knots and links using a special property derived from their minimal diagrams. A knot or link K′ is called a predecessor of a knot or link K if Cr(K′) < Cr(K) and a diagram of K′ can be obtained from a minimal diagram D of K by a single crossing change. In such a case we say that K′ < K. We investigate the sets of knots that can be obtain...

Knots from Combinatorics of Knot Diagrams to Combinatorial Topology Based on Knots

The Kontsevich Integral and Algebraic Structures on the Space of Diagrams

This paper is part expository and part presentation of calculational results. The target space of the Kontsevich integral for knots is a space of diagrams; this space has various algebraic structures which are described here. These are utilized with Le’s theorem on the behaviour of the Kontsevich integral under cabling and with the Melvin-Morton Theorem, to obtain, in the Kontsevich integral fo...

Mirror-curve Codes for Knots and Links

Inspired by Lomonaco–Kauffman paper on quantum knots and knot mosaics we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. We introduce codes for mirror-curves treated as knot or link diagrams placed in rectangular square grid, suitable for software implementations and discuss possible applications of mirror-curves.