An AJ conjecture for trivalent graphs

par Ramanujan Santharoubane (Orsay)

vendredi 2 oct. 2026, 14:00 → 15:00 Europe/Paris

Salle Fokko du Cloux (Bat. Braconnier)

The coloured Jones polynomial is a sequence of Laurent polynomials associated with a knot. In 2003, Garoufalidis and Lê proved that the coloured Jones polynomial of a given knot satisfies a kind of recurrence relation, called a
q-difference equation. One year later, Garoufalidis proposed a conjecture, now known as the AJ conjecture, stating that a certain evaluation of this q-difference equation recovers the A-polynomial of the knot. The A-polynomial is an important classical invariant: a two-variable Laurent polynomial that encodes the restriction map from the SL(2,C) character variety of the knot exterior to that of its boundary torus.

In this talk I will explain how to generalise this conjecture to embeddings of trivalent graphs in the 3-sphere, replacing the coloured Jones polynomial by the coloured Kauffman bracket of the graph. This generalisation recovers the original conjecture, and we will see that for planar embeddings the new conjecture holds.

This is joint work in progress with Renaud Detcherry and Louis Ioos.