Abstract
We prove the meridional rank conjecture for twisted links and arborescent links associated to bipartite trees with even weights. These links are substantial generalizations of pretzels and two-bridge links, respectively. Lower bounds on meridional rank are obtained via Coxeter quotients of the groups of link complements. Matching upper bounds on bridge number are found using the Wirtinger numbers of link diagrams, a combinatorial tool developed by the authors.
Highlights
The meridional rank μ of a link L in S3 is the minimal number of meridians of L needed to generate π1(S3\L). It is an immediate consequence of the Wirtinger presentation for π1(S3\L) in a suitable diagram that μ(L) is bounded above by the bridge number β(L)
We prove the meridional rank conjecture for two new classes, twisted links and arborescent links associated with bipartite trees with even weights
A a b b c c a b c c b d a a b b d c a Theorem 2 The meridional rank conjecture holds for arborescent links associated with bipartite trees with non-zero even weights
Summary
The meridional rank μ of a link L in S3 is the minimal number of meridians of L needed to generate π1(S3\L). We prove the meridional rank conjecture for two new classes, twisted links and arborescent links associated with bipartite trees with even weights. The class of arborescent links generalizes both two-bridge links and Montesinos links They are defined by plumbing twisted bands in a tree-like pattern. The class of arborescent links associated with even weight bipartite trees contains all two-bridge links. The class of arborescent links associated with even weight bipartite trees contains the class of slalom divide links defined by A’Campo [1] These links are obtained by plumbing positive Hopf bands along bipartite trees. Theorem 2 The meridional rank conjecture holds for arborescent links associated with bipartite trees with non-zero even weights. Additivity of meridional rank under connected sum, which is implied by the meridional rank conjecture, is an interesting open question in its own right
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