Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » The Art of the PFUG » Legendrian knots and links

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Rice Digital Scholarship

    This collection is included in aLens by: Digital Scholarship at Rice University

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

Also in these lenses

  • Lens for Engineering

    This collection is included inLens: Lens for Engineering
    By: Sidney Burrus

    Click the "Lens for Engineering" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.
 

Legendrian knots and links

Module by: Prudence Heck. E-mail the author

Summary: 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 R3R3. In this PFUG we ask the same question when restrictions are placed on the way the sticks may lay in R3R3. 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 ξξ on R3R3 assigns to each point pR3pR3 a plane ξpR3ξpR3. The standard contact structure assigns to the point p=(x,y,z)p=(x,y,z) the plane

ξ p = Span { e 2 , e 1 + y e 3 } , ξ p = Span { e 2 , e 1 + y e 3 } ,
(1)

where we orient R3R3 via the Right Hand Rule.

Figure 1: The standard contact structure. Figure courtesy of the Wikipedia article on contact geometry, http://en.wikipedia.org/wiki/Contact_geometry.
Figure 1 (contact1.png)

Definition 2.2 A knot in R3R3 is the image of an embedding Φ:S1R3Φ:S1R3. A Legendrian knot in R3,ξR3,ξ is a knot such that

Φ ' ( θ ) ξ Φ ( θ ) Φ ' ( θ ) ξ Φ ( θ )
(2)

for all θθ.

Observe that if Φ(θ)=x(θ),y(θ),z(θ)Φ(θ)=x(θ),y(θ),z(θ) parametrizes a knot LL then for LL to be Legendrian in the standard contact structure, Φ'(θ)Φ'(θ) must be orthogonal to e2×(e1+ye3)=ye1-e3. In particular, y(θ)x'(θ)-z'(θ)=0 for all θ .e2×(e1+ye3)=ye1-e3. In particular, y(θ)x'(θ)-z'(θ)=0 for all θ .

Whenever we draw a knot in R3R3 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.

Figure 2: A Legendrian unknot in the front projection, left, and Lagrangian projection, right.
Figure 2 (Knots2.png)

Definition 2.3 Let Π:R3R2Π:R3R2 such that Π(x,y,z)=(x,z)Π(x,y,z)=(x,z). The front projection of LL, denoted Π(L)Π(L), is the image of LL under ΠΠ. If ΦΦ above parametrizes LL then ΦΠ(θ):=x(θ),z(θ)ΦΠ(θ):=x(θ),z(θ) parametrizes Π(L)Π(L).

We see from the identity y(θ)x'(θ)-z'(θ)=0y(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:R3R2π:R3R2 such that π(x,y,z)=(x,y)π(x,y,z)=(x,y). The Lagrangian projection of LL is the image of LL under ππ and is denoted π(L)π(L). As with the front projection, π(L)π(L) is parametrized by Φπ(θ):=x(θ),y(θ)Φπ(θ):=x(θ),y(θ).

Reidemeister Moves

Definition 3.1 Two Legendrian knots are Legendrian isotopic if there is an isotopy H:S1×IR3H:S1×IR3 between them such that Ht:S1R3Ht:S1R3 parametrizes a Legendrian knot for each tItI.

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),andΩ(3),Ω(1),Ω(2),andΩ(3), below.

Figure 3: Reidemeister Moves for the front projection.
Figure 3 (FrontRmoves1.png)

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

Figure 4: Reidemeister Moves for the Lagrangian projection.
Figure 4 (LagRmoves1.png)

Remark 3.4 The converse to Theorem 3.3 is false: there exist non-Legendrian isotopic Legendrian knots whose Lagrangian projections are related by a finite sequence of the moves σ(1)σ(1) and σ(2)σ(2).

Invariants of Legendrian Knots

Definition 4.1 The rotation number of an oriented Legendrian knot is

r ( L ) = 1 2 ( D - U ) , r ( L ) = 1 2 ( D - U ) ,
(3)

where DD is the number of down cusps in the front projection and UU is the number of up cusps.

Definition 4.2 The Thurston-Bennequin invariant, tb(L)tb(L), is the linking number between LL and L'L', where L'L' is a slight push-off of LL in the zz-direction.

That is, tb(L)tb(L) is one-half the signed count of the intersections between LL and L'L', where we assign ±1±1 to intersections as in the following diagram:

Figure 5: Following the Right Hand Rule, a positive crossing, left, and negative crossing, right.
Figure 5 (Crossings.png)

Theorem 4.3 Two oriented Legendrian torus knots are Legendrian isotopic if and only if their Thurston-Bennequin invariants, rotation numbers, and (topological) knot types agree.

Remark 4.4 Theorem 4.3 is not true for all knots. Y. Chekanov and Y. Eliashberg independently gave examples of non-Legendrian isotopic Legendrian knots with the same Thurston-Bennequin invariant, rotation number, and topological knot type.

Stick Number

Definition 5.1 A stick knot is the image of a PL-embedding of S1S1 in R3R3. That is, a knot composed of line segments attached to each other at their endpoints. We call the segments edges and the joined endpoints vertices, and we require that exactly two edges meet at each vertex. The stick number S(L)S(L) of a knot LL is the minimum number of sticks necessary to make LL in R3R3.

Definition 5.2 A Legendrian stick knot is a stick knot that is Legendrian everywhere except at its vertices. The Legendrian stick number of a Legendrian knot LL, denoted LS(L)LS(L), is the minimum number of sticks necessary to make a stick knot that is Legendrian isotopic to LL away from its vertices. The front stick number of LL, FS(L)FS(L), is the minimum number of sticks in a front diagram of LL.

Remark 5.3 It is not hard to see that LS(L)=2FS(L)LS(L)=2FS(L). For, if a stick lies in ξξ then it is parametrized by a line segment

l ( t ) = x ( t ) , y ( t ) , z ( t ) = ( m x t + x 0 , m y t + y 0 , m z t + z 0 ) l ( t ) = x ( t ) , y ( t ) , z ( t ) = ( m x t + x 0 , m y t + y 0 , m z t + z 0 )
(4)

for tItI such that y(t)x'(t)-z'(t)=0y(t)x'(t)-z'(t)=0. In particular, either mx=mz=0mx=mz=0, in which case the stick runs perpendicular to the xzxz-plane, or my=0my=0, and the stick lies in a plane parallel to the xzxz-plane.

Example 5.4 M M

  1. a.: The minimum FSFS number over all Legendrian unknots is 3.
  2. b.: The Legendrian right-handed trefoil TRTR with tb=1tb=1 has FS(TR)=6FS(TR)=6. The (topological) right-handed trefoil has S(TR)=6S(TR)=6.

Questions

  1. Define isotopy for Legendrian stick knots.
  2. Define Reidemeister moves for Legendrian stick knots in the front projection. Is there an analogue of Theorem 3.2?
  3. Are there analogues of the Thurston-Bennequin invariant and rotation number for Legendrian stick knots? How do they behave under the Reidemeister moves?
  4. Is there some relationship between FS(L)FS(L) and S(L)S(L)?
  5. In an REU with J. Sabloff, J. Ralston and S. Sacchetti showed that for the maximal Thurston-Bennequin representative of the torus knot T(2,2n-1)T(2,2n-1), FST(2,2n-1)2n+4FST(2,2n-1)2n+4. Can this be improved? Is there a bound for general torus knots?

Acknowledgments

This Connexions module describes work conducted as part of Rice University's VIGRE program, supported by National Science Foundation grant DMS-0739420. We would like to thank Dr. Josh Sabloff for suggesting this problem on Legendrian sticks, Dr. Shelly Harvey for assisting in the organization of our PFUG, and the undergraduate members C. Buenger, A. Jamshidi, S. Kruzick, A. Mehta, and M. Scherf.

References

  1. Adams, Colin C. and Brennan, Bevin M. and Greilsheimer, Deborah L. and Woo, Alexander K. (1997). Stick numbers and composition of knots and links. J. Knot Theory Ramifications, 6(2), 149-161.
  2. Adams, Colin C. (2004). The knot book. (Revised reprint of the 1994 original). Providence, RI: American Mathematical Society.
  3. Etnyre, John. Legendrian and Transversal Knots. [To Appear in the Handbook of Knot Theory].
  4. Livingston, Charles. (1993). Carus Mathematical Monographs: Vol. 24. Knot theory. (). Washington, DC: Mathematical Association of America.
  5. Sabloff, Joshua. Invariants for Legendrian knots from contact homology. [Preprint].
  6. Sabloff, Joshua. (2009, August). Personal correspondence.

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks