Rectangle condition and its applications

Abstract: In this paper, we define the rectangle condition on the bridge sphere for a

n

-bridge decomposition of a knot whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of

3

-manifolds. We show that the satisfaction of the rectangle condition for a

n

-bridge decomposition can guarantee that the Hempel distance for the

n

-bridge decomposition is greater than or equal to

2

. In particular, we give an interesting family of alternating 3-bridge knots by using the rectangle condition and a modified train track argument.

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