A family of Andrews–Curtis trivializations via 4-manifold trisections

An R-link is an n-component link L in \(S^3\) $S^3$ such that Dehn surgery on L yields \(\#^n(S^1 \times S^2)\) ${\#}^n(S^1\times S^2)$ . Every R-link L gives rise to a geometrically simply-connected homotopy 4-sphere \(X_L\) $X_L$ , which in turn can be used to produce a balanced presentation of the trivial group. Adapting work of Gompf, Scharlemann, and Thompson, Meier and Zupan 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, \(L(3,2;2n/(2n+1))\) $L(3,2;2n/(2n+1))$ induces the infamous trivial group presentation \(\langle x,y \, | \, xyx=yxy, x^{n+1}=y^n \rangle \) $⟨x,y|xyx=yxy,x^{n+1}=y^n⟩$ , a popular collection of potential counterexamples to the Andrews–Curtis conjecture for \(n \ge 3\) $n\ge 3$ . In this paper, we 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.