1. SIAM Conf. on Math for Industry, Oct. 10, 2009 Modeling Knots for Aesthetics and Simulations Carlo H. Séquin U.C. Berkeley Modeling, Analysis, Design …

2. Knots in Clothing

3. Knotted Appliances Garden hose Power cable

4. Intricate Knots in the Realm of... Boats Horses

5. Knots in Art Macrame Sculpture

6. Knotted Plants Kelp Lianas

7. Knotted Building Blocks of Life Knotted DNA Model of the most complex knotted protein (MIT 2006)

8. 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 unknot

9. Various Unknots

10. 3D Hilbert Curve ( Séquin 2006 )

11. Pax Mundi II ( 2007 ) Brent Collins, Steve Reinmuth, Carlo Séquin

12. The Simplest Real Knot: The Trefoil José de Rivera, Construction #35 M. C. Escher, Knots (1965)

13. Complex, Symmetrical Knots

14. Tight “Braided” Knots

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

16. Links and Linked Knots A link: comprises a set of loops – possibly knotted and tangled together.

17. Two Linked Tori: Link 2 2 1 John Robinson, Bonds of Friendship (1979)

18. Borromean Rings: Link 6 3 2 John Robinson

19. Tetra Trefoil Tangles Simple linking (1) -- Complex linking (2) {over-over-under-under} {over-under-over-under}

20. Tetrahedral Trefoil Tangle (FDM)

21. A Loose Tangle of Trefoils

22. Dodecahedral Pentafoil Cluster

23. Realization: Extrude Hone - ProMetal Metal sintering and infiltration process

24. A Split Trefoil To open: Rotate around z-axis

25. Split Trefoil (side view, closed)

26. Split Trefoil (side view, open)

27. Splitting Moebius Bands Litho by FDM-model FDM-model M.C.Escher thin, colored thick

28. Split Moebius Trefoil (Séquin, 2003)

29. “Knot Divided” Breckenridge, 2005

30. Knotty Problem How many crossings does this “Not-Divided” Knot have ?

31. 2.5D Celtic Knots – Basic Step

32. Celtic Knot – Denser Configuration

33. Celtic Knot – Second Iteration

34. Recursive 9-Crossing Knot Is this really a 81-crossing knot ? 9 crossings

35. 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.

36. Project: “Beauty of Knots” Find maximal symmetry in 3D for simple knots. Knot 4 1 and Knot 6 1

37. Computer Representation of Knots Spline representation via its control polygon. String of piecewise-linear line segments. But...

38. Is the Control Polygon Representative? A Problem: You may construct a nice knotted control polygon, and then find that the spline curve it defines is not knotted at all !

39. Unknot With Knotted Control-Polygon Composite of two cubic Bézier curves

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

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

42. An Intriguing Question: Can an un-knotted control polygon produce a knotted spline curve ? First guess: Probably NOT Variation-diminishing property of Bézier curves implies that a spline cannot “wiggle” more than its control polygon.

43. Cubic Bézier and Its Control Polygon Cubic Bézier curve Region where curve is “outside” of control polygon Two “entangled” curves With “non-entangled” control polygons Convex hull of control polygon

44. Two “Entangled” Bezier Segments “in 3D” NOTE: The 2 control polygons are NOT entangled!

45. The Building Block Two “entangled” curves With “non-entangled” control polygons

46. Combining 4 such Entangled Units Use several units …

47. Control Polygons Are NOT Entangled … Use several units …

48. Can Be Reduced to the Chords

49. This Is NOT a Knot !

50. But This Is a Knot ! Knot 7 2

51. The Problem When can we use the control polygon to make reliable predictions about the curve ? Thus we have a true spline knot whose control polygon is the unknot !

52. Tubular Neighborhoods A tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle. (Wikipedia) ( Tom Peters et al.)

53. Ambient Isotopy If both the curve and its control polygon lie in the same tubular neighborhood, they have the same topological surroundings and thus have the same knotted-ness. ( Tom Peters et al.) subdivided control polygon

54. A “Safe” Tubular Neighborhood A tube of uniform diameter equal to the minimum separation of any two branches

55. More Efficient Neighborhoods? Tube diameter is determined by tightest bottleneck Inefficient! Make tube diameter variable along the knot curve(s) Difficult!

56. Another “Neighborhood” The notion of the “control ribbon”: control polygon spline curve control ribbon A ruled surface, that connects points with equal parameter values on the spline and on the control polygon

57. Knots and Their Control Ribbons K3 1 : “Trefoil” and K9 40 : “Chinese Button Knot”

58. Crucial Test on Control Ribbon Any self-intersections ?

59. Does a Line Pass thru Control Ribbon? Look at the “crossings” formed by close approaches between query line (green) and the edges of the control ribbon. If the two “crossings” have the same sign, line stabs the ribbon.

60. Current Focus Find out how this can be done most efficiently: u Find the occurrences of all “close approaches” u Determine the signs of the relevant “crossings”

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

62. 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)).

63. Q U E S T I O N S ? Granny-Knot-Lattice (Squin, 1981) Granny-Knot-Lattice (Séquin, 1981)