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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 …. Knots in Clothing . Knotted Appliances . Garden hose Power cable. Intricate Knots in the Realm of . . .

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siam conf on math for industry oct 10 2009
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
Knotted Appliances
  • Garden hose Power cable
knots in art
Knots in Art
  • Macrame Sculpture
knotted plants
Knotted Plants
  • Kelp Lianas
knotted building blocks of life
Knotted Building Blocks of Life
  • Knotted DNA

Model of the most complex knotted protein (MIT 2006)

mathematicians knots
Mathematicians’ Knots


  • 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
Pax Mundi II (2007)
  • Brent Collins, Steve Reinmuth, Carlo Séquin
the simplest real knot the trefoil
The Simplest Real Knot: The Trefoil
  • José de Rivera, Construction #35

M. C. Escher, Knots (1965)

composite knots
Composite Knots
  • Knots can be “opened” at their periphery and then connected to each other.
links and linked knots
Links and Linked Knots
  • A link: comprises a set of loops
  • – possibly knotted and tangled together.
two linked tori link 2 2 1
Two Linked Tori: Link 221

John Robinson, Bonds of Friendship (1979)

tetra trefoil tangles
Tetra Trefoil Tangles
  • Simple linking (1) -- Complex linking (2)
  • {over-over-under-under} {over-under-over-under}
realization extrude hone prometal
Realization: Extrude Hone - ProMetal
  • Metal sintering and infiltration process
a split trefoil
A Split Trefoil
  • To open: Rotate around z-axis
splitting moebius bands
Splitting Moebius Bands
  • Litho by FDM-model FDM-modelM.C.Escher thin, colored thick
knotty problem
Knotty Problem
  • How many crossings
  • does this “Not-Divided” Knot have ?
recursive 9 crossing knot
Recursive 9-Crossing Knot

9 crossings

  • Is this really a 81-crossing knot ?
knot classification
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.
project beauty of knots
Project: “Beauty of Knots”
  • Find maximal symmetry in 3D for simple knots.

Knot 41 and Knot 61

computer representation of knots
Computer Representation of Knots

String of piecewise-linear line segments.

  • Spline representation via its control polygon.

But . . .

is the control polygon representative
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
Unknot With Knotted Control-Polygon
  • Composite of two cubic Bézier curves
highly knotted control polygons
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 polygons1
Highly Knotted Control-Polygons
  • The Result:
  • Control polygon has 12 crossings.
  • Compound Bézier curve is still the unknot!
an intriguing question
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 b zier and its control polygon
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
Two “Entangled” Bezier Segments “in 3D”
  • NOTE: The 2 control polygons are NOT entangled!
the building block
The Building Block

Two “entangled” curves

With “non-entangled” control polygons

the problem
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
Tubular Neighborhoods

( Tom Peters et al.)


  • A tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.
ambient isotopy
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 “Safe” Tubular Neighborhood
  • A tube of uniform diameter equal to the minimum separation of any two branches
more efficient neighborhoods
More Efficient Neighborhoods?

Make tube diametervariable along the knot curve(s)


Tube diameter is determined by tightest bottleneck


another neighborhood
Another “Neighborhood”

control polygon


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
Knots and Their Control Ribbons
  • K31: “Trefoil” and K940: “Chinese Button Knot”
crucial test on control ribbon
Crucial Test on Control Ribbon
  • Any self-intersections ?
does a line pass thru control ribbon
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
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”
  • 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!
  • 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
Q U E S T I O N S ?

Granny-Knot-Lattice (Séquin, 1981)