## – Europe/Lisbon

Online

João Faria Martins, University of Leeds

Crossed modules, homotopy 2-types, knotted surfaces and welded knots

I will review the construction of invariants of knots, loop braids and knotted surfaces derived from finite crossed modules. I will also show a method to calculate the algebraic homotopy 2-type of the complement of a knotted surface $\Sigma$ embedded in the 4-sphere from a movie presentation of $\Sigma$. This will entail a categorified form of the Wirtinger relations for a knot group. Along the way I will also show applications to welded knots in terms of a biquandle related to the homotopy 2-type of the complement of the tube of a welded knots.

The last stages of this talk are part of the framework of the Leverhulme Trust research project grant: RPG-2018-029: *Emergent Physics From Lattice Models of Higher Gauge Theory*.

### Additional file

Projecto FCT `UIDB/04459/2020`.