2023/04/06 by Romary, Ethan, Zupan, Alexander
#20F05 #57K40 #57R60 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2304.03196
An R-link is an n-component link L in S3 such that Dehn surgery on L yields #n(S1 × S2). Every R-link L gives rise to a geometrically simply-connected homotopy 4-sphere XL, 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 ⟨ x,y | xyx=yxy, xn+1=yn ⟩, a popular collection of potential counterexamples to the Andrews-Curtis conjecture for n ≥ 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.