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Thurston's geometrization conjecture and its subsequent proof
for Haken manifolds distinguish knots in S^3 by the geometries in the
complement of the knots. While the definition of alternating knots make
use of nice knot diagrams, Knot Floer homology, a knot invariant
toolbox, defined by Ozsvath-Szabo and Rasumussen, generalizes the
definition of alternating knots in the context of knot Floer homology
and defines family of quasi-alternating knots which contains all
alternating knots. Using Lipshitz-Ozsvath-Thurston's bordered Floer
homology, we prove a partial affirmation of a folklore conjecture in
knot Floer theory, which bridges these two viewpoints of looking at
knots. |