Celebratio Mathematica — Rourke — Complete Bibliography

Complete Bibliography

Works connected to Sofia S. F. Lambropoulou

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S. Lambropoulou and C. P. Rourke: “Markov’s theorem in 3-manifolds,” Topology Appl. 78 : 1–2 (1997), pp. 95–122. MR 1465027 Zbl 0879.57007 article

S. Lambropoulou and C. P. Rourke: “Algebraic Markov equivalence for links in three-manifolds,” Compos. Math. 142 : 4 (2006), pp. 1039–1062. MR 2249541 Zbl 1156.57007 article

Let \( B_n \) denote the classical braid group on \( n \) strands and let the mixed braid group \( B_{m,n} \) be the subgroup of \( B_{m+n} \) comprising braids for which the first \( m \) strands form the identity braid. Let \[ B_{m,\infty}=\bigcup_nB_{m,n} .\] We describe explicit algebraic moves on \( B_{m,\infty} \) such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented three-manifold. The moves depend on a fixed link representing the manifold in \( S^3 \). More precisely, for link complements the moves are the two familiar moves of the classical Markov equivalence together with ‘twisted’ conjugation by certain loops \( a_i \). This means premultiplication by \( a_i^{-1} \) and postmultiplication by a ‘combed’ version of \( a_i \). For closed three-manifolds there is an additional set of ‘combed’ band moves that correspond to sliding moves over the surgery link. The main tool in the proofs is the one-move Markov theorem using \( L \)-moves (adding in-box crossings). The resulting algebraic classification is a direct extension of the classical Markov theorem that classifies links in \( S^3 \) up to isotopy, and potentially leads to powerful new link invariants, which have been explored in special cases by the first author. It also provides a controlled range of isotopy moves, useful for studying skein modules of three-manifolds.