The Andrews-Curtis conjecture asserts that “any balanced presentation \(\langle x_1,\dots,x_n \mid r_1,\dots,r_n\rangle \) of the trivial group can be converted to the trivial presentation \(\langle x_1,\dots,x_n \mid x_1,\dots,x_n\rangle \) by a finite sequence of the following moves:
1. Replace a relator \(r_i\) by \(r_i^{-1}\)
2. Replace a relator \(r_i\) by \(r_ir_j\) where \(i \neq j\)
3. Replace a relator \(r_i\) by \(x_jr_ix_j^{-1}\)
4. Add or delete a trivial generator/relator pair \(x_{n+1}\) and \(r_{n+1} = x_{n+1}\)
A presentation \(P\) that admits such a trivialization is called AC-trivial. Although the conjecture remains open, there are interesting families of potential counterexamples, many arising from constructions in low-dimensional topology.”
“An R-link is an \(n\)-component link \(L\) in \(S^3\) such that Dehn surgery on \(L\) yields \(\#^n(S^1 \times S^2)\). Every R-link \(L\) gives rise to a geometrically simply-connected homotopy 4-sphere \(X_L\), which in turn can be used to produce a balanced presentation of the trivial group. Adapting work of Gompf, Scharlemann, and Thompson [R. E. Gompf et al., Geom. Topol. 14, No. 4, 2305–2347 (2010; Zbl 1214.57008)], J. Meier and A. Zupan [J. Differ. Geom. 122, No. 1, 69–129 (2022; Zbl 1521.57017)] produced a family of R-links \(L(p,q;c/d)\), where the pairs \((p,q)\) and \((c,d)\) are relatively prime and \(c\) is even. Within this family, \(L(3, 2; 2n/(2n + 1))\) induces the trivial group presentation \(\langle x, y \mid xyx = yxy, x^{n+1} = y^n\rangle \), a popular collection of potential counterexamples to the Andrews-Curtis conjecture for \(n \geq 3\).”
In this paper, the authors use 4-manifold trisections to show that the group presentations corresponding to a different subfamily, \(L(3,2;4/d)\), are Andrews-Curtis trivial for all \(d\).