The long-eight-figure spines of lens spaces and binary trees (Q1868191)

The notion of complexity \(c(M)\) of a 3-manifold \(M\) is due to \textit{S. V. Matveev} [Acta Appl. Math. 19, 101-130 (1990; Zbl 0724.57012)]: \(c(M)\) is defined to be the minimum number of vertices of an almost special spine \(P\) for \(M.\) For each lens space \(L_{p,q}\), \(p\geq 3\), the almost special spine \(P_{p,q}\), called the long-eight-figure spine, is known; moreover, the equality \[ (*) \qquad c(L_{p,q})= c(P_{p,q}) \] has been conjectured to hold (see Matveev's tables of 3-manifolds up to complexity 6, yielding a proof via computer calculation, in case \(c(P_{p,q})\leq 8\)). The present paper deals with a problem strictly related to attempts to prove \((*)\), i.e. the identification of pre-images of the embedded spine \(P_{p,q}\) with respect to the natural combinatorial presentation of lens space \(L_{p,q}.\) The main theorem makes use of the one-to-one correspondence between the set of lens spaces with \(p>2\) and the set of non-empty binary codes, up to inversions and reversions, and of the so called unzip move on binary trees associated to binary codes. In case \(p>3\), the result of the work appears to be a method to resolve the singularity of the spine preserving the embedding in \(L_{p,q}\); so, the obtained spine allows to reconstruct the lens space unambigously (see the quoted paper by Matveev).