Bridge number, Heegaard genus and non-integral Dehn surgery

We show there exists a linear function w: N->N with the following property. Let K be a hyperbolic knot in a hyperbolic 3-manifold M admitting a non-longitudinal S^3 surgery. If K is put into thin position with respect to a strongly irreducible, genus g Heegaard splitting of M then K intersects a thick level at most 2w(g) times. Typically, this shows that the bridge number of K with respect to this Heegaard splitting is at most w(g), and the tunnel number of K is at most w(g) + g-1.Comment: 76 page, 48 figures; referee comments incorporated and typos fixed; accepted at TAM

The formation of mixed germanium–cobalt carbonyl clusters: an electrospray mass spectrometric study, and the structure of a high-nuclearity [Ge₂Co₁₀(CO)₂₄]²⁻ anion

The reaction of [µ₄-Ge{Co₂(CO)₇}₂] with [Co(CO)₄]⁻ has been monitored by electrospray mass spectrometry to detect the cluster anions generated. Conditions giving known mixed Ge–Co carbonyl clusters were established, and a new high nuclearity cluster anion, [Ge₂Co₁₀(CO)₂₄]²⁻ was detected. Conditions for its formation were optimised and it was subsequently isolated as its [Et₄N]⁺ salt and characterised by single-crystal X-ray crystallography. The Ge₂Co₁₀ cluster core has a novel geometry with the two germanium atoms in semi-encapsulated positions, forming seven formal Ge–Co bonds. There are also eighteen formal Co–Co bonds. Corresponding reactions of [µ₄-Si{Co₂(CO)₇}₂] with [Co(CO)₄]⁻ were also investigated