Legendrian knots and links

Introduction

In studying knots, mathematicians often make the assumption that all knots under consideration are smooth. Yet knots that appear in nature are often rigid. A stick knot, or polygonal knot, is a knot composed of line segments attached at their edges. Much work has been done over the last few decades towards determining the minimum number of sticks necessary to construct a given knot in ${\mathbb{R}}^3$ . In this PFUG we ask the same question when restrictions are placed on the way the sticks may lay in ${\mathbb{R}}^3$ . That is, we seek to determine the minimum number of Legendrian sticks necessary to construct a given Legendrian knot.

Preliminaries

Definition 2.1 A contact structure $\xi$ on ${\mathbb{R}}^3$ assigns to each point $p\in {\mathbb{R}}^3$ a plane ${\xi }_p\subset {\mathbb{R}}^3$ . The standard contact structure assigns to the point $p=(x,y,z)$ the plane

where we orient ${\mathbb{R}}^3$ via the Right Hand Rule.

Definition 2.2 A knot in ${\mathbb{R}}^3$ is the image of an embedding $\Phi :S^1\to {\mathbb{R}}^3$ . A Legendrian knot in $\left({\mathbb{R}}^3,,,\xi \right)$ is a knot such that

for all $\theta$ .

Observe that if $\Phi \left(\theta \right)=\left(x,\left(,\theta ,\right),,,y,\left(,\theta ,\right),,,z,\left(,\theta ,\right)\right)$ parametrizes a knot $L$ then for $L$ to be Legendrian in the standard contact structure, ${\Phi }^{\text{'}}\left(\theta \right)$ must be orthogonal to ${\overrightarrow{e}}_2\times ({\overrightarrow{e}}_1+y{\overrightarrow{e}}_3)=y{\overrightarrow{e}}_1-{\overrightarrow{e}}_3\text{. In particular,}y\left(\theta \right)x^{\text{'}}\left(\theta \right)-z^{\text{'}}\left(\theta \right)=0\text{for all}\mathrm{\theta \; .}$

Whenever we draw a knot in ${\mathbb{R}}^3$ we are actually drawing a projection of the knot in some plane with labeled crossings. We use the front and Lagrangian projections to draw Legendrian knots.

Definition 2.3 Let $\Pi :{\mathbb{R}}^3\to {\mathbb{R}}^2$ such that $\Pi (x,y,z)=(x,z)$ . The front projection of $L$ , denoted $\Pi \left(L\right)$ , is the image of $L$ under $\Pi$ . If $\Phi$ above parametrizes $L$ then ${\Phi }_{\Pi }\left(\theta \right):=\left(x,\left(,\theta ,\right),,,z,\left(,\theta ,\right)\right)$ parametrizes $\Pi \left(L\right)$ .

We see from the identity $y\left(\theta \right)x^{\text{'}}\left(\theta \right)-z^{\text{'}}\left(\theta \right)=0$ that the front projection of a Legendrian knot cannot have vertical tangencies, and at each crossing the slope of the over-crossing segment must be less than the slope of the under-crossing segment.

Definition 2.4 Let $\pi :{\mathbb{R}}^3\to {\mathbb{R}}^2$ such that $\pi (x,y,z)=(x,y)$ . The Lagrangian projection of $L$ is the image of $L$ under $\pi$ and is denoted $\pi \left(L\right)$ . As with the front projection, $\pi \left(L\right)$ is parametrized by ${\Phi }_{\pi }\left(\theta \right):=\left(x,\left(,\theta ,\right),,,y,\left(,\theta ,\right)\right)$ .

Reidemeister moves

Definition 3.1 Two Legendrian knots are Legendrian isotopic if there is an isotopy $H:S^1\times I\to {\mathbb{R}}^3$ between them such that $H_t:S^1\to {\mathbb{R}}^3$ parametrizes a Legendrian knot for each $t\in I$ .

Theorem 3.2 Two front projections represent Legendrian isotopic Legendrian knots if and only if they are related by regular isotopy and a finite sequence of the moves $\Omega \left(1\right),\Omega \left(2\right),\text{and}\Omega \left(3\right),$ below.

Theorem 3.3 If two Lagrangian projections represent Legendrian isotopic Legendrian knots then they are related by a sequence of the moves $\sigma \left(1\right)$ and $\sigma \left(2\right)$ , below.