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

Abstract

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, 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. For this family, $L(3,2;2n/(2n+1))$ induces the infamous trivial group presentation $\langle x,y \, | \, xyx=yxy, x^{n+1}=y^n \rangle$, a popular collection of potential counterexamples to the Andrews-Curtis conjecture for $n \geq 3$. In this paper, we use 4-manifold trisections to show that the group presentations corresponding to a different family, $L(3,2;4/d)$, are Andrews-Curtis trivial for all $d$.