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Bridge Positions and Plat Presentations of Links

Seth Hovland

Abstract

In this paper we investigate the relationship between links in bridge position and plat presentations. We will show that the Hilden double coset classes of plat presentations of a link are equivalent to bridge positions of the link up to bridge isotopy. This correspondence allows us to reframe algebraic questions about plat presentations in terms of bridge positions. We demonstrate some results about both plat presentations and links in bridge position using this correspondence. For instance, we reprove that there is only one Hilden double coset class of the n-bridge unknot in $\mathbb{S}^3.$ We also show that there is only a single double coset class for torus knots in plat position. Finally, we discuss how this correspondence may be used to investigate plat closures of knots, which is the subject of ongoing research.

Bridge Positions and Plat Presentations of Links

Abstract

In this paper we investigate the relationship between links in bridge position and plat presentations. We will show that the Hilden double coset classes of plat presentations of a link are equivalent to bridge positions of the link up to bridge isotopy. This correspondence allows us to reframe algebraic questions about plat presentations in terms of bridge positions. We demonstrate some results about both plat presentations and links in bridge position using this correspondence. For instance, we reprove that there is only one Hilden double coset class of the n-bridge unknot in We also show that there is only a single double coset class for torus knots in plat position. Finally, we discuss how this correspondence may be used to investigate plat closures of knots, which is the subject of ongoing research.

Paper Structure

This paper contains 19 sections, 18 theorems, 1 equation, 20 figures.

Table of Contents

  1. Introduction
  2. Bridge Position of a Knot
  3. Bridge Positions and the Suspension of $S^2$
  4. Plat Presentation of a Knot
  5. Plats and Double Cosets
  6. Reidemeister Moves on Plats and the Hilden Subgroup
  7. The Pocket Move
  8. Alternative View of Plat Positions
  9. Plat Equivalence is Bridge Isotopy
  10. Proof using Alternative View of Bridge and Plat Positions
  11. Applications of Equivalence
  12. The Hilden Double Coset Problem
  13. Summary of Results about Hilden Double Coset Classes of Special Knots
  14. Discussion of Further Directions
  15. A Hilden Double Coset Invariant
  16. ...and 4 more sections

Key Result

Theorem 2.5

Two links are topologically equivalent if and only if any two bridge positions can be related by a sequence of stabilizations and destabilizations and bridge isotopy.

Figures (20)

  • Figure 1: The height function on $S^2$ induced from z-projection. Notice pre-images of regular values are $S^1$'s
  • Figure 2: An arbitary link in bridge position with a stabilization/destabilization occuring along the red subarc
  • Figure 3: Suspension of $S^2.$ Left: The outer red sphere and inner blue sphere are contracted to a single point. Right: Removing a "core" from the picture on the left we obtain a solid cube the top and bottom face are contracted to a single point.
  • Figure 4: A knot in bridge position with respect to height function from projecting onto the $I$ value in the suspension of $S^2.$
  • Figure 5: A Bridge isotopy realized as isotoping level spheres. The first figure is of a knot in bridge position and the $t=t_1$ sphere outlined in green. This sphere is then isotoped to push up the point where the lower bridge touches tangent to the $t=t_1$ sphere, the result on the knot is that the lower bridge is now raised, however intersections of the knot on the level spheres remains unchanged!
  • ...and 15 more figures

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Example 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • Definition 3.5
  • ...and 23 more