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## SIAM Conf. on Math for Industry, Oct. 10, 2009

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SIAM Conf. on Math for Industry, Oct. 10, 2009

Carlo H. Séquin

U.C. Berkeley

- Modeling Knots for Aesthetics and Simulations

Modeling, Analysis, Design …

Knotted Appliances

- Garden hose Power cable

Intricate Knots in the Realm of . . .

- Boats Horses

Knots in Art

- Macrame Sculpture

Knotted Plants

- Kelp Lianas

Mathematicians’ Knots

unknot

- Closed, non-self-intersecting curves in 3D space

0 3 4 6

Tabulated by their crossing-number :

= The minimal number of crossings visible after any deformation and projection

Pax Mundi II (2007)

- Brent Collins, Steve Reinmuth, Carlo Séquin

Composite Knots

- Knots can be “opened” at their periphery and then connected to each other.

Links and Linked Knots

- A link: comprises a set of loops
- – possibly knotted and tangled together.

Two Linked Tori: Link 221

John Robinson, Bonds of Friendship (1979)

Borromean Rings: Link 632

John Robinson

Tetra Trefoil Tangles

- Simple linking (1) -- Complex linking (2)
- {over-over-under-under} {over-under-over-under}

Realization: Extrude Hone - ProMetal

- Metal sintering and infiltration process

A Split Trefoil

- To open: Rotate around z-axis

Splitting Moebius Bands

- Litho by FDM-model FDM-modelM.C.Escher thin, colored thick

Knotty Problem

- How many crossings
- does this “Not-Divided” Knot have ?

Knot Classification

- What kind of knot is this ?
- Can you just look it up in the knot tables ?
- How do you find a projection that yields the minimum number of crossings ?
- There is still no completely safe method to assure that two knots are the same.

Computer Representation of Knots

String of piecewise-linear line segments.

- Spline representation via its control polygon.

But . . .

Is the Control Polygon Representative?

You may construct a nice knotted control polygon,and then find that the spline curve it defines is not knotted at all !

- A Problem:

Unknot With Knotted Control-Polygon

- Composite of two cubic Bézier curves

Highly Knotted Control-Polygons

- Use the previous configuration as a building block.
- Cut open lower left joint between the 2 Bézier segments.
- Small changes will keep the control polygons knotted.
- Assemble several such constructs in a cyclic compound.

Highly Knotted Control-Polygons

- The Result:
- Control polygon has 12 crossings.
- Compound Bézier curve is still the unknot!

An Intriguing Question:

First guess: Probably NOT

Variation-diminishing property of Bézier curves

implies that a spline cannot “wiggle”

more than its control polygon.

- Can an un-knotted control polygon
- produce a knotted spline curve ?

Cubic Bézier and Its Control Polygon

Two “entangled” curves

With “non-entangled” control polygons

Convex hull of control polygon

Region where curve is “outside” of control polygon

Cubic Bézier curve

Two “Entangled” Bezier Segments “in 3D”

- NOTE: The 2 control polygons are NOT entangled!

Combining 4 such Entangled Units

- Use several units …

Control Polygons Are NOT Entangled …

- Use several units …

But This Is a Knot !

Knot 72

The Problem

Thus we have a true spline knot

whose control polygon is the unknot !

- When can we use the control polygon to make reliable predictions about the curve ?

Tubular Neighborhoods

( Tom Peters et al.)

(Wikipedia)

- A tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.

Ambient Isotopy

( Tom Peters et al.)

- If both the curve and its control polygon lie in the same tubular neighborhood,they have the same topological surroundingsand thus have the same knotted-ness.

subdivided control polygon

A “Safe” Tubular Neighborhood

- A tube of uniform diameter equal to the minimum separation of any two branches

More Efficient Neighborhoods?

Make tube diametervariable along the knot curve(s)

Difficult!

Tube diameter is determined by tightest bottleneck

Inefficient!

Another “Neighborhood”

control polygon

controlribbon

spline curve

- The notion of the “control ribbon”:

A ruled surface, that connects points with equal parameter values on the spline and on the control polygon

Knots and Their Control Ribbons

- K31: “Trefoil” and K940: “Chinese Button Knot”

Crucial Test on Control Ribbon

- Any self-intersections ?

Does a Line Pass thru Control Ribbon?

- Look at the “crossings”formed by close approaches betweenquery line (green)and the edges ofthe control ribbon.
- If the two “crossings” have the same sign,line stabs the ribbon.

Current Focus

- Find out how this can be done most efficiently:
- Find the occurrences of all “close approaches”
- Determine the signs of the relevant “crossings”

Conclusion

- Knots appear in many domains, in many different forms, and with highly varying degrees of complexity.
- CAD tools have only tangentially addressed efficient modeling and analysis of knotted structures.
- Suitable abstractions of knots, coupled with some topological guarantees, offer promise for computationally efficient solutions.
- The “quest” has only just begun!

Acknowledgements

- Thanks to Tom Peters for many fruitful discussions!
- This work is being supported in part by the Center for Hybrid and Embedded Software Systems (CHESS) at UC Berkeley, which receives support from the National Science Foundation (NSF award #CCR-0225610 (ITR)).

Q U E S T I O N S ?

Granny-Knot-Lattice (Séquin, 1981)

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