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Abstract
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Isotopy classes of circles on an orientable surface
of genus
form a
quandle
under the operation of Dehn twisting about such circles. We derive certain
fundamental relations in the Dehn quandle and then consider a homology theory
based on this quandle. We show how certain types of relations in the quandle
translate into cycles and homology representatives in this homology theory, and
characterize a large family of 2–cycles representing homology elements. Finally we
draw connections to Lefschetz fibrations, showing isomorphism classes of such
fibrations over a disk correspond to quandle homology classes in dimension 2, and
discuss some further structures on the homology.
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Keywords
quandle homology, Dehn twist, Lefschetz fibration
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Mathematical Subject Classification 2000
Primary: 18G60, 57T99
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Publication
Received: 4 October 2007
Accepted: 22 October 2007
Published: 8 February 2008
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