Upper Bounds for the Complexity of Torus Knot Complements

Abstract

We establish upper bounds for the complexity of Seifert fibered manifolds with nonempty boundary. In particular, we obtain potentially sharp bounds on the complexity of torus knot complements.

Computing Matveev's complexity via crystallization theory: the boundary case

“…While writing the present paper, we learnt about a preprint by Fominykh and Wiest, which also yields an upper estimation for Matveev's complexity of torus knot complements, via their representation as Seifert manifolds: see[18]. In some cases, Fominykh and Wiest's estimation further improves Matveev's one.…”

The complexity of orientable graph manifolds

“…Let θ p k /q k be the theta graph in Θ p k /q k that is closest to θ + . The skeleton X k for Φ k is obtained by assembling several skeletons of type P F connecting the theta graph P ∩ Φ k to a theta graph which is one step closer to θ + than θ p j /q j ,with respect to the distance on Θ(T 2 ) (see [6]). The number of the required flips is either S(p j , q j ) − 2 or S(p j , q j ) − 1 depending on the shift chosen for the corresponding component of A used in the construction of the skeleton P .…”

“…Upper bounds for the complexity of infinite families of 3-manifolds are given in [11] for lens spaces, in [10] for closed orientable Seifert fibre spaces and for orientable torus bundles over the circle, in [6] for orientable Sefert fibre space with boundary and in [2] for non-orientable compact Seifert fibre spaces. All the previous upper bounds are sharp for manifolds contained in the above cited catalogues.…”

Three-dimensional manifolds with poor spines

“…At present, the exact values of complexity are known only for a finite number of tabulated manifolds [2,3,4] and for several infinite families of manifolds with boundary [5,6,7,8], closed manifolds [9,10], and cusped manifolds [11,12]. Estimates for the complexity of some infinite families of manifolds were obtained in [13,14,15,16,17].…”