Browsing by Author "Brandenbursky, Michael"

Now showing 1 - 3 of 3
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    Invariants of closed braids via counting surfaces
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2012) Brandenbursky, Michael
    A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram associated with a closed braid. We then identify the resulting invariants with partial derivatives of the HOMFLY-PT polynomial.
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    On the autonomous metric on the group of area-preserving diffeomorphisms of the 2-disc
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2012) Brandenbursky, Michael
    Let D2 be the open unit disc in the Euclidean plane and let G := Diff(D2; area) be the group of smooth compactly supported area-preserving diffeomorphisms of D2. For every natural number k we construct an injective homomorphism Zk → G, which is bi-Lipschitz with respect to the word metric on Zk and the autonomous metric on G. We also show that the space of homogeneous quasi-morphisms vanishing on all autonomous diffeomorphisms in the above group is infinite dimensional.
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    On the autonomous metric on the groups of Hamiltonian diffeomorphisms of closed hyperbolic surfaces
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Brandenbursky, Michael
    Let g be a closed hyperbolic surface of genus g and let Ham g be the group of Hamiltonian diffeomorphisms of g. The most natural word metric on this group is the autonomous metric. It has many interesting properties, most important of which is the bi-invariance of this metric. In this work we show that Ham g is unbounded with respect to this metric