The Faithfulness Question for the Burau Representation
نویسندگان
چکیده
منابع مشابه
The Faithfulness Question for the Burau Representation
We consider the Burau and Gassner representations of the braid groups B„ . A sufficient condition for faithfulness, involving just a pair of arcs, is shown to be necessary as well for all but at most two values of n. In the Burau case, this implies nonfaithfulness for n > 10 . Alexander showed in [6] that any knot K can be made by connecting the ends of some braid b. The Alexander polynomial A(...
متن کاملWeak Faithfulness Properties for the Burau Representation
We study the components of the matrices that belong to the image of the Burau representation of braids, and establish both algebraic and order constraints for a given Laurent polynomial possibly be a component of such a Burau matrix. As an application partial faithfulness results for the Burau representation are deduced. AMS Classification: 20F36, 20H25, 15A24, 57M05.
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The topological entropy of a braid is the infimum of the entropies of all homeomor-phisms of the disc which have a finite invariant set represented by the braid. When the isotopy class represented by the braid is pseudo-Anosov or is reducible with a pseudo-Anosov component, this entropy is positive. Fried and Kolev proved that the entropy is bounded below by the logarithm of the spectral radius...
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The Burau representation enables to define many other representations of the braid group Bn by the topological operation of “cabling braids”. We show here that these representations split into copies of the Burau representation itself and of a representation of Bn/(Pn, Pn). In particular, we show that there is no gain in terms of faithfulness by cabling the Burau representation.
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1993
ISSN: 0002-9939
DOI: 10.2307/2159956