<< Chapter < Page Chapter >> Page >
VIGRE is a program sponsored by the National Science Foundation to carry out innovative educational programs in which research and education are integrated and in which undergraduates, graduate students, postdoctoral fellows, and faculty are mutually supportive. This work outlines the Topology PFUG on Legendrian knots and links offered at Rice University as MATH 499, section 004, in the Fall of 2009.

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 R 3 . In this PFUG we ask the same question when restrictions are placed on the way the sticks may lay in 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 ξ on R 3 assigns to each point p R 3 a plane ξ p R 3 . The standard contact structure assigns to the point p = ( x , y , z ) the plane

ξ p = Span { e 2 , e 1 + y e 3 } ,

where we orient R 3 via the Right Hand Rule.

The standard contact structure. Figure courtesy of the Wikipedia article on contact geometry, http://en.wikipedia.org/wiki/Contact_geometry.

Definition 2.2 A knot in R 3 is the image of an embedding Φ : S 1 R 3 . A Legendrian knot in R 3 , ξ is a knot such that

Φ ' ( θ ) ξ Φ ( θ )

for all θ .

Observe that if Φ ( θ ) = x ( θ ) , y ( θ ) , z ( θ ) parametrizes a knot L then for L to be Legendrian in the standard contact structure, Φ ' ( θ ) must be orthogonal to e 2 × ( e 1 + y e 3 ) = y e 1 - e 3 . In particular, y ( θ ) x ' ( θ ) - z ' ( θ ) = 0 for all θ .

Whenever we draw a knot in 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.

A Legendrian unknot in the front projection, left, and Lagrangian projection, right.

Definition 2.3 Let Π : R 3 R 2 such that Π ( x , y , z ) = ( x , z ) . The front projection of L , denoted Π ( L ) , is the image of L under Π . If Φ above parametrizes L then Φ Π ( θ ) : = x ( θ ) , z ( θ ) parametrizes Π ( L ) .

We see from the identity y ( θ ) x ' ( θ ) - z ' ( θ ) = 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 π : R 3 R 2 such that π ( x , y , z ) = ( x , y ) . The Lagrangian projection of L is the image of L under π and is denoted π ( L ) . As with the front projection, π ( L ) is parametrized by Φ π ( θ ) : = x ( θ ) , y ( θ ) .

Reidemeister moves

Definition 3.1 Two Legendrian knots are Legendrian isotopic if there is an isotopy H : S 1 × I R 3 between them such that H t : S 1 R 3 parametrizes a Legendrian knot for each t 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 Ω ( 1 ) , Ω ( 2 ) , and Ω ( 3 ) , below.

Reidemeister Moves for the front projection.

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

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The art of the pfug' conversation and receive update notifications?

Ask