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

A Taste of Differential Geometry: Ribbons and White's FormulaA Taste of Differential Geometry: Ribbons and White’s Formula
Richard G. Ligo
I Linking number I Writhe I Twist
I White’s Theorem
Tangent vector to the “Pringles” curve:
P(t) =
For example:
A calculation at t = π:
P(π) =
U =
A calculation at t = π:
P(π) =
U =
The fruit of our calculations:
White’s Formula
Richard G. Ligo
White’s Formula
Richard G. Ligo
Ribbon definition
A curve and normal vector field along it define a ribbon.
White’s Formula
Richard G. Ligo
Ribbon definition
A curve and normal vector field along it define a ribbon.
White’s Formula
Richard G. Ligo
A local crossing occurs when the ribbon appears “edge-on.”
Conversely, a nonlocal crossing occurs when one section passes over another.
White’s Formula
Richard G. Ligo
White’s Formula
Richard G. Ligo
To calculate the linking number, assign crossing values to the crossings between the two edges, compute their sum, and divide by two.
Lk(R) = −1+1+1+1 2 = 2
2 = 1
Intuitively, Lk(R) describes the “tangledness” of the ribbon edges of R.
White’s Formula
Richard G. Ligo
The writhe
To calculate the writhe, choose a ribbon edge, assign crossing values to its self-crossings, and compute their sum.
Wr(R) = +1 = 1
Intuitively, Wr(R) describes the “nonplanarity” of the chosen ribbon edge.
White’s Formula
Richard G. Ligo
The twist
To calculate the twist, assign crossing values to the local crossings, compute their sum, and divide by two.
Tw(R) = −1+1 2 = 0
2 = 0
Intuitively, Tw(R) describes how much the ribbon edges rotate about each other.
White’s Formula
Richard G. Ligo
Lk(R) = 1 Wr(R) = 1 Tw(R) = 0
Lk(R) = Wr(R) + Tw(R)
White’s Formula
Richard G. Ligo
Lk(R) = 1 Wr(R) = 1 Tw(R) = 0
Lk(R) = Wr(R) + Tw(R)
White’s Formula
Richard G. Ligo
Lk(R) = 1 Wr(R) = 1 Tw(R) = 0
Lk(R) = Wr(R) + Tw(R)
White’s Formula
Richard G. Ligo
Another example:
Lk(R) = −1+1+1+1+1+1+1+1 2 = 6
2 = 3
Another example:
Lk(R) = −1+1+1+1+1+1+1+1 2 = 6
2 = 3
2 = 1
Lk(R) = Wr(R) + Tw(R)
White’s Formula
Richard G. Ligo
White’s Formula
Richard G. Ligo
White’s Formula
Richard G. Ligo
2 = 1
2 = 1
Lk(R) = Wr(R) + Tw(R)
White’s Formula
Richard G. Ligo
The Gauss-Bonnet Theorem
Like White’s Formula, the Gauss-Bonnet Theorem illustrates a connection between topological and geometric properties.
Theorem: If M is a compact, two-dimensional, Riemannian manifold with boundary ∂M, then∫
M K dA +
kg ds = 2πχ(M),
Where K is the Gaussian curvature on M, kg is the geodesic curvature on ∂M, and χ(M) is the Euler characteristic of M.
White’s Formula
Richard G. Ligo
The Gauss-Bonnet Theorem
Like White’s Formula, the Gauss-Bonnet Theorem illustrates a connection between topological and geometric properties.
Theorem: If M is a compact, two-dimensional, Riemannian manifold with boundary ∂M, then∫
M K dA +
kg ds = 2πχ(M),
Where K is the Gaussian curvature on M, kg is the geodesic curvature on ∂M, and χ(M) is the Euler characteristic of M.
White’s Formula
Richard G. Ligo
So, what can you do with this?
Direct applications:
White’s Formula
Richard G. Ligo
So, what can you do with this?
Direct applications:
White’s Formula
Richard G. Ligo
Energy stored within a ribbon is given by the integral
E = 1
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
Energy stored within a ribbon is given by the integral
E = 1
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
Energy stored within a ribbon is given by the integral
E = 1
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
I Space curves can be used to define ribbons.
I The behavior of ribbons is constrained by White’s formula.
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 predict configurations.
White’s Formula
Richard G. Ligo
I Can we model curves with nonuniform density?
I Is it possible to obtain results from purely geometrical methods?
White’s Formula
Richard G. Ligo
I The University of Iowa
References:
I M. Dennis & J. Hannay. The geometry of Calugareanu’s Theorem.
I J. Hearst & Y. Shi. The Kirchoff elastic rod, the nonlinear Shrodinger equations, and DNA supercoiling.
I K. Hu. Writhe of DNA induced by a terminal twist.
I M. Podvratnik. Torsional instability of elastic rods.
Introduction
Definitions
Examples
Application
Conclusion