UGA international

Linkext is a strategic research collaboration between Department of Mathematics, and Institut Fourier. Launched in 2025 as a laureate of the NTU-UGA-UT Trilateral Center, this project focuses on a new generalization of link-homotopy, a central notion in low dimensional topology introduced in the fifties by John Milnor (Fields medal 1962). More precisely, the central idea is to extend the notion of link-homotopy from links in the 3-sphere, to links embedded in the complement of a fixed link F. More precisely, for a given link F, our goal is to study in details the set LH(n;F) of n-component links considered up to link-homotopy within the complement of F.

Tangible Results

The first key step of the project is to show that this generalized notion of link-homotopy is natural and well motivated. When F is empty, it reduces to the classical link-homotopy. When n=1, LH(1;F) coincides with the fundamental group of the complement of F, linking the construction to classical algebraic topology. Thus, LH(n;F) is not merely formal but grounded in established theory.
Initial progress has clarified the structure of LH(n;F) in simple cases. For n=2 with F the trivial knot, the situation already shows unexpected complexity. There are infinitely many distinct classes that do not appear in the classical case, detectable via refined Milnor invariants, showing that even a trivial F deeply changes the classification.
In parallel, the project studies algebraic invariants from quandles. Building on work by Ishii, Kotorii, and Murao, these invariants extend to LH(n;F) and may reveal features beyond Milnor invariants, providing complementary insight.

Testimonials from principal investigators 

This project offers a unique opportunity for our doctoral researchers to benefit from a rich, multicultural research environment. Atsushii ISHII Associate Professor - Department of Mathematics (University of Tsukuba, Japan) View researcher’s profile

The project allowed to give life to a completely new and promising collaboration, where our two fields of expertise can meet in a complementary way. Jean-Baptiste MEILHAN Associate Professor (Université Grenoble Alpes, France) Institut Fourier View researcher’s profile