After graduation, Apollos began a Ph.D. program in mathematics at Louisiana State University (LSU).
The central breakthrough of the paper is the explicit construction of a new linear representation of the n-braid group. While classical rack invariants work well for closed links, they fail for open braids because the rack operation is deterministic, making the raw counting invariant trivial. To overcome this, the authors introduce a "pointed rack," an algebraic structure equipped with basepoints to remember the specific color assignments at the top and bottom of a braid diagram.
This framework allows them to formalize a set-theoretic permutation representation of the n-braid group acting on the space of rack colorings. By doing so, the abstract topological action of concatenating braids is successfully translated into the concrete, algebraic operation of matrix multiplication. As a direct byproduct of establishing this linear representation, the authors introduce a new matrix-valued invariant.
Rather than taking the trace of the linear operator, which collapses the structural data into a single integer, preserving the full "rack counting matrix" retains the rich, off-diagonal information that tracks exactly how individual colorings propagate from top to bottom.
Explicit computational examples confirm that this uncollapsed matrix representation successfully distinguishes topological differences in open braids that older, trace-based invariants treat as identical.
Posted August 21, 2026
