Bridge distance and plat projections

Jesse Johnson; Yoav Moriah

doi:10.2140/agt.2016.16.3361

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arXiv · 1312.7093

Bridge distance and plat projections

Jesse Johnson·Yoav Moriah

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Abstract

We calculate the bridge distance for $m$-bridge knots/links in the $3$-sphere with sufficiently complicated $2m$-plat projections. In particular we show that if the underlying braid of the plat has $n - 1$ rows of twists and all its exponents have absolute value greater than or equal to three then the distance of the bridge sphere is exactly $\lceil n/(2(m - 2)) \rceil$, where $\lceil x \rceil$ is the smallest integer greater than or equal to $x$. As a corollary, we conclude that if such a diagram has more than $4m(m-2)$ rows then the bridge sphere defining the plat projection is the unique minimal bridge sphere for the knot.

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Jesse Johnson, Yoav Moriah. 2013-12-26. Bridge distance and plat projections. https://doi.org/10.2140/agt.2016.16.3361

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