A Taste of Differential Geometry: Ribbons and White's Formula

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

A Taste of Differential Geometry:Ribbons and White’s Formula

Richard G. Ligo

The University of Iowa

April 15, 2015

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

A common task

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

Overview

I IntroductionI Definitions

I Space curvesI Ribbons

I Linking numberI WritheI Twist

I White’s Theorem

I Examples

I Application

I Conclusion

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

Space curves

The “Pringles” curve:

P(t) =

cos(t)sin(t)

sin(t) cos(t)

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

The tangent to a curve

Tangent vector to the “Pringles” curve:

P(t) =

cos(t)sin(t)

sin(t) cos(t)

=⇒ T (t) =

− sin(t)cos(t)

cos2(t)− sin2(t)

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

A normal vector to a curve

For example:

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion


A calculation at t = π:

P(π) =

cos(π)sin(π)

sin(π) cos(π)

=

−100

T (π) =

− sin(π)cos(π)

cos2(π)− sin2(π)

=

0−11

We can then spot a normal vector:

U =

011

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion


The fruit of our calculations:

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

A unit normal vector field to a curve

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

Ribbon definition

A curve and normal vector field along it define a ribbon.

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

Crossing types

A local crossing occurs when the ribbon appears “edge-on.”

Conversely, a nonlocal crossing occurs when one sectionpasses over another.

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

Crossing values

Positive crossings Negative crossings

Positive crossings are assigned a value of +1.

Negative crossings are assigned a value of −1.

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

The linking number

To calculate the linking number, assign crossing values tothe crossings between the two edges, compute their sum,and divide by two.

Lk(R) = −1+1+1+12 = 2

2 = 1

Intuitively, Lk(R) describes the “tangledness” of the ribbonedges of R.

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

The writhe

To calculate the writhe, choose a ribbon edge, assigncrossing values to its self-crossings, and compute their sum.

Wr(R) = +1 = 1

Intuitively, Wr(R) describes the “nonplanarity” of the chosenribbon edge.

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

The twist

To calculate the twist, assign crossing values to the localcrossings, compute their sum, and divide by two.

Tw(R) = −1+12 = 0

2 = 0

Intuitively, Tw(R) describes how much the ribbon edgesrotate about each other.

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

White’s formula

White’s formula connects Lk, Wr, and Tw.

Lk(R) = 1 Wr(R) = 1 Tw(R) = 0

Lk(R) = Wr(R) + Tw(R)

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

White’s formula

Another example:

Lk(R) = −1+1+1+1+1+1+1+12 = 6

2 = 3

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

White’s formula

Another example:

Wr(R) = +1 + 1 = 2

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

White’s formula

Another example:

Tw(R) = −1+1+1+12 = 2

2 = 1

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

White’s formula

Another example:

Lk(R) = 3 Wr(R) = 2 Tw(R) = 1


White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

White’s formula

Apply an isotopy:

Lk(R) is invariant, but Wr(R) and Tw(R) can change.

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

White’s formula

After isotopy:

Lk(R) = +1+12 = 2

2 = 1

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

White’s formula

After isotopy:

Wr(R) = 0

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

White’s formula

After isotopy:

Tw(R) = +1+12 = 2

2 = 1

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

White’s formula

After isotopy:

Lk(R) = 1 Wr(R) = 0 Tw(R) = 1


White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

The Gauss-Bonnet Theorem

Like White’s Formula, the Gauss-Bonnet Theorem illustratesa connection between topological and geometric properties.

Theorem: If M is a compact, two-dimensional, Riemannianmanifold with boundary ∂M, then∫

MK dA +

∫∂M

kg ds = 2πχ(M),

Where K is the Gaussian curvature on M, kg is the geodesiccurvature on ∂M, and χ(M) is the Euler characteristic of M.

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

Returning to the extension cord...

So, what can you do with this?

Direct applications:

I Engineering (modeling cable, wire, rope, etc.)

I Biology (modeling DNA packing, DNA transcription)

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

Energy minimization

Energy stored within a ribbon is given by the integral

E =1

2

∫ `

0aκ2 + bτ2 dt.

κ is the curvature at a point.

τ is the torsion at a point.

low κ and τ high κ, low τ low κ, high τ high κ and τ

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

Possible configurations

Circle Helix

Plectoneme

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

Conclusion

Process summary:

I Space curves can be used to define ribbons.

I The behavior of ribbons is constrained by White’sformula.

I Ribbons can be used to model real-life situations.

I Real-life situations desire to minimize stored energy.

I Methods from differential geometry allow us to predictconfigurations.

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

Conclusion

Open questions:

I How do we model plectonemic regions?

I Can we model curves with nonuniform density?

I Is it possible to obtain results from purely geometricalmethods?

White’s Formula

Richard G. Ligo

Introduction

Definitions

Examples

Application

Conclusion

Conclusion

Acknowledgements:

I Dr. Oguz Durumeric

I G. Calugareanu, B. Fuller, and J. H. White

I The University of Iowa

References:

I M. Dennis & J. Hannay. The geometry ofCalugareanu’s Theorem.

I J. Hearst & Y. Shi. The Kirchoff elastic rod, thenonlinear Shrodinger equations, and DNA supercoiling.

I K. Hu. Writhe of DNA induced by a terminal twist.

I M. Podvratnik. Torsional instability of elastic rods.

A Taste of Differential Geometry: Ribbons and White's Formula

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