Invariants of $2$-knots

TL;DR

This survey reviews various algebraic, gauge theoretic, and combinatorial invariants of 2-knots in 4-dimensional spheres, highlighting the current state of knowledge without presenting new results.

Contribution

It provides a concise overview of existing invariants of 2-knots, summarizing their algebraic and geometric foundations without introducing new findings.

Findings

Summarizes algebraic invariants like Alexander and Casson-Gordon invariants.

Discusses gauge theoretic and combinatorial invariants for 2-knots.

Highlights the scarcity of detailed results and new discoveries in the field.

Abstract

This short survey, which was written to accompany a minicourse at the BIRS conference "Topology in dimension 4.5", concerns invariants of knotted $2$-spheres in $S^4$, also known as $2$-knots. It covers invariants extracted from the algebraic topology of the knot exterior, including Alexander invariants, the Farber-Levine pairing and Casson-Gordon invariants, as well as gauge theoretic and combinatorial invariants. Details are scarce and new results inexistant.