Arrow calculus for welded links

Akira Yasuhara (Waseda University)

We develop a calculus for diagrams of knotted objects. We define arrow presentations, which encode the crossing information of a diagram into arrows in a way somewhat similar to Gauss diagrams, and more generally w–tree presentations, which can be seen as “higher-order Gauss diagrams”. This arrow calculus is used to develop an analogue of Habiro’s clasper theory for welded knotted objects, which contain classical link diagrams as a subset. This provides a “realization” of Polyak’s algebra of arrow diagrams at the welded level, and leads to a characterization of finite- type invariants of welded knots and long knots. This is a joint work with Jean-Baptiste Meilhan (University of Grenoble Alpes).

The reductivity of a knot projection

Ayaka Shimizu (NIT, Gunma College, Japan)

The reductivity of a knot projection is defined to be the minimum number of splices required to make the projection reducible, where the splices are applied to a knot projection resulting in another knot projection. It has been shown that the reductivity is four or less for any knot projection and shown that there are infinitely many knot projections with reductivity 0, 1, 2, and 3. The “reductivity problem” is a problem asking the existence of a knot projection whose reductivity is four. In this talk, we will discuss some strategies for the reductivity problem focusing on the region of a knot projection.

Slides for talk: