Motivic Knots and the Abhyankar-Moh Conjecture

The Abhyankar-Moh theorem in affine algebraic geometry states that any polynomial embedding i:\mathbb{C}\hookrightarrow\mathbb{C}^3 can be rectified. This means there exists a polynomial automorphism f of \mathbb{C}^3 such that f\circ i = t \mapsto (t, 0, 0).
The Abhyankar-Moh conjecture generalizes this idea: It proposes that any polynomial embedding \mathbb{C}^k\hookrightarrow\mathbb{C}^n can be rectified, for all dimensions k and n. While this is known to hold when n > 2k + 1, the conjecture remains open even for specific cases. For example, it is unsolved for the embedding \mathbb{C}\hookrightarrow\mathbb{C}^3 : t \mapsto (t^3-3t, t^4-4t^2, t^5-10t).
A promising approach to this conjecture uses techniques from geometric topology, especially knot theory. Recent research explores how Morel-Voevodsky’s motivic homotopy theory can bridge topological methods and algebraic geometry, offering new strategies for such problems.
In this talk, I will overview the basics of modern algebraic geometry and motivic homotopy theory. The goal is to introduce motivic knots and their invariants. Familiarity with commutative algebra and category theory is assumed.