Bridge distance and plat projections

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.Comment: 21 pages 13 figure

An infinite family of links with critical bridge spheres

“…Our construction is inspired by recent work of Johnson and Moriah [6] in which they construct links with bridge surfaces of arbitrarily high distance. The central result of this paper is the following: Theorem 6.9.…”

“…Following that, in Section 3 we embed a link L in S 3 in a plat position with bridge sphere F and discuss some of the specific details of the embedding as well as build some of the tools (certain arcs, loops, disks, and projection maps) which we will use throughout the rest of the paper. Sections 2 and 3 should be considered setup for the rest of the paper, and this is essentially the same setup as in Johnson and Moriah's paper [6].…”

An infinite family of links with critical bridge spheres

“…Our construction is inspired by recent work of Johnson and Moriah [6] in which they construct links with bridge surfaces of arbitrarily high distance. The central result of this thesis is the following:…”

“…Following that, in Section 3 we embed a link L in S 3 in a plat position with bridge sphere F and discuss some of the specific details of the embedding as well as build some of the tools (certain arcs, loops, disks, and projection maps) which we will use throughout the rest of the thesis. Sections 2 and 3 should be considered setup for the rest of the thesis, and this is essentially the same setup as in Johnson and Moriah's paper [6]. In particular we make use of Johnson and Moriah's plat links and projection maps.…”

“…τ -projection: We will also need a more subtle type of projection which respects L. We closely follow Johnson and Moriah [6]: Note that L intersects each F i in the same number of points and these points vary continuously as i varies from 1 to h.…”