## Interactive Topological Drawing (1998)

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Citations: | 18 - 1 self |

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@TECHREPORT{Scharein98interactivetopological,

author = {Robert Glenn Scharein},

title = {Interactive Topological Drawing},

institution = {},

year = {1998}

}

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### Abstract

The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in three-dimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with the correct representation and display of these objects on a computer. By correctness we mean that the essential topological features of objects are maintained during interaction. We have chosen to limit the scope of topological drawing to knot theory, a domain that consists essentially of one class of object (embedded circles in three-dimensional space) yet is rich enough to contain a wide variety of difficult problems of research interest. In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or self-intersections. This notion of equivalence may be thought of as the heart of knot theory. We present methods for the computer construction and interactive manipulation of a

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Citation Context ...may be roughly divided into approaches that use gradient-descent on the energy landscape, and are therefore deterministic, and those which employ non-deterministic methods such as simulated annealing =-=[KGV84]-=- or other methods of random perturbation. The deterministic approach is in some ways more appealing, since it would be extremely interesting to discover that gradient-descent on a suitable knot energy... |

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Citation Context ... really one. 3.5.3 Lorenz knots The Lorenz equations are a system of first-order differential equations dx dt = σ(y − x), dy dt = x(r − z) − y, dz dt = xy − bz that were originally proposed by Lorenz =-=[Lor63]-=- in a model of ocean currents. These equations are a simple example of a dynamical system that exhibits chaos [Jac90]. Figure 3.32 shows the flow on R 3 for the Lorenz equations starting at several di... |

664 | Surface reconstruction from unorganized points - Hoppe, DeRose, et al. - 1992 |

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Citation Context ...t is currently installed at more than two dozen institutions and is being actively used as a research or visualization tool [KBM + 96, MSS97, Sed96, KOP + 97, BSS98b, CDW98]. A planned port to OpenGL =-=[NDW93]-=- and Java will allow the program to run on many less costly personal computers. This will greatly increase the program’s availability, given the rapidly improving graphics performance of these machine... |

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Citation Context ...gether are called a link. Each constituent knot of the link is known as a component of the link. To simplify the terminology in the thesis, we will often refer to links as knots and, following Conway =-=[Con70]-=-, use the term proper knot when specifically referring to one-component knots. Two knots or links are equivalent if there exists an orientation-preserving homeomorphism on R 3 that maps one knot onto ... |

117 |
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Citation Context ...r, the 2-twist spun knot is distinct from the ordinary (or 0-twist) spun knot. More complex deformations than mere twisting can be performed which yield still more complicated knots in R 4 . Kawauchi =-=[Kaw96]-=- gives a good exposition as well as the paper by Roseman [Ros89]. 57sChapter 4 KnotPlot KnotPlot is a program for visualizing and interacting with 3D and 4D knots. Its features are far too numerous to... |

113 |
A study in interactive 3-D rotation using 2-D control device
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(Show Context)
Citation Context ...hysical realism to them, and all are easily mastered in a matter of moments. In the final analysis it probably doesn’t matter mathematically, as there is only one algebra of 3D rotations. Chen’s work =-=[CMS88]-=- is apparently the first implementation of a virtual trackball in the literature. It is no doubt wide-spread by now. Implementing it from the description in the paper can be tricky because of a number... |

91 |
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(Show Context)
Citation Context ...ch90]). This unknot makes its appearance in this thesis at the end of Chapter 7. Kusner and Sullivan [KS94] relax a discrete version of the conformal energy. They use Brakke’s Surface Evolver program =-=[Bra92]-=- to perform conjugate gradient descent on the energy. They find this method is effective at untangling unknots such as the Freedman unknot (Figure 3.15). Kusner and Sullivan produce a table of energy-... |

86 |
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(Show Context)
Citation Context ...smooth transition or metamorphosis between two different objects). Although most researchers in the area are well aware of the difficulties of morphing between two arbitrary three-dimensional objects =-=[Hug92]-=-, DeCarlo and Gallier [DG96] point out that many do not take topology seriously. The fact that all the power of modern mathematics is yet unable to effectively “handle” cases such as Figure 1.1 indica... |

83 | A user interface for specifying three-dimensional orientation using a mouse
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(Show Context)
Citation Context ... that leaves that space. There has been some discussion in the literature about which formulation of a 3D trackball is the most intuitive or useful, or the most efficient at performing specific tasks =-=[Sho92]-=-. It seems that this is largely a matter of personal taste and the degree of training and familiarity with each system. All of the trackballs described in this section appear to be within “error limit... |

80 |
Geometrical Methods of Mathematical Physics
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(Show Context)
Citation Context ...ted in a natural way because of the lack of commutativity inherent in 3D rotations. This means that a sequence of rotations AB is in general not undone by the sequence A −1 B −1 . According to Schutz =-=[Sch90]-=- this lack of commutativity is necessary if infinitesimal operations in a two-dimensional space are to produce a net result that leaves that space. There has been some discussion in the literature abo... |

78 |
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Citation Context ...am into the Laurent polynomials 9 that is invariant under the Reidemeister moves. Alexander originally derived the polynomial that bears his name by a method completely independent of skein relations =-=[Ale28]-=-. The techniques he used are outside the scope of this thesis, however, consult [Rol76, BZ85] to read about the important mathematical rôle played by the Alexander polynomial (among several significan... |

73 |
The Philosophy of Space and Time
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Citation Context ...alization, the reader would be advised to seek out the paper by Roseman [Ros95] and the references it contains. Other excellent sources are the books by Hilbert and Cohn-Vossen [HCV52] or Reichenbach =-=[Rei57]-=-. The collection of articles edited by Brisson [Bri78b] are “early” works in the field of mathematical visualization by computer. 1.3 Contributions of the thesis “Your work is interesting; it’s not co... |

64 |
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(Show Context)
Citation Context ... x), dy dt = x(r − z) − y, dz dt = xy − bz that were originally proposed by Lorenz [Lor63] in a model of ocean currents. These equations are a simple example of a dynamical system that exhibits chaos =-=[Jac90]-=-. Figure 3.32 shows the flow on R 3 for the Lorenz equations starting at several different positions (indicated by a dot). 17 17 In Figure 3.32 the initial positions all lie on the attractor; empirica... |

62 | Knots and links in spatial graphs - Gordon - 1983 |

60 |
On systems of knotted curves
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(Show Context)
Citation Context ...rators and relations form the braid group Bn on n-strings. These groups have many fascinating properties. One further connection of braid theory with knot theory comes from a theorem due to Alexander =-=[Ale23]-=- that any knot or link may be represented as a closed braid. Abraidmaybe closed by connecting the top ends to the bottom ends, in sequence, in such a way so as not to A B Figure 3.29: Composition of b... |

55 | The computational complexity of knot and link problems
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(Show Context)
Citation Context ...g Reidemeister move. principle [Hak61, Hem92], is only known to have an algorithm of such high complexity that it is useless for all practical purposes. Recent results of Haas, Lagarias and Pippenger =-=[HLP97]-=- show that the problem of detecting unknottedness is in the complexity class NP. This is “good news” in that, although still difficult, the problem is easier than had hitherto been believed. The diffi... |

53 | On the total curvature of knots - Milnor - 1950 |

52 | Theorie der Normalflächen - Haken - 1961 |

49 |
On the energy of knots and unknots
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(Show Context)
Citation Context ... be made more complex before it can be simplified (where complexity is defined as the number of crossings in the diagram). The “monster” from Figure 1.1 as well as the “Freedman (un)knot” (taken from =-=[FHW94]-=-) in Figure 3.15 cannot be simplified by either one of the first two Reidemeister moves (Figure 3.3), both of which reduce the number of crossings. In fact, as will be shown in Chapter 7, the first of... |

45 | Ocneanu A new polynomial invariants of Knots and Links - Freyd, Yetter, et al. - 1985 |

45 |
The first 1,701,936 knots
- Hoste, Thistlethwaite, et al.
- 1998
(Show Context)
Citation Context ... due to using computer techniques and efficient means of encoding knot diagrams. Hoste, Thistlethwaite, and Weeks have now determined that there are 1,701,935 prime knots 6 with 16 or fewer crossings =-=[HTW98]-=-. To check results, the researchers worked in two independent teams using completely different methods, none-the-less coming up with the same list of knots. In Chapter 6 we will return to the discussi... |

44 |
A polynomial invariant for links via Von Neumann algebras
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(Show Context)
Citation Context ... of these, an 11-crossing knot, is shown in Figure 3.24. For most of the history of knot theory, the Alexander polynomial was the only knot polynomial. It was quite a shock to many when Vaughan Jones =-=[Jon85]-=- discovered a powerful new polynomial invariant for links in 1985 defined by the skein relation 1 VL+ (t) − tVL− (t) = t Vunknot(t) = 1 � √t � 1 − √t VL0 (t) The Jones polynomial is far more discrimin... |

42 |
A quick trip through knot theory
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(Show Context)
Citation Context .... The arena chosen for demonstrating topological drawing is knot theory, for several reasons. First of all, knot theory can be considered to be the simplest, nontrivial example of a placement problem =-=[Fox62]-=-. Other placement problems (motion control in robotics, human body movement) inherit some of the difficulties already found in knot theory. Secondly, although knot theory is a beautiful branch of mode... |

38 |
Geomview: An interactive geometry viewer
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(Show Context)
Citation Context ...s ability to navigate through elliptic or hyperbolic space and to illuminate surfaces there. Perhaps the premier mathematical visualization software package anywhere is the Geometry Center’s GeomView =-=[PLM93]-=-. GeomView, like Salem, can display objects in spherical or hyperbolic space. GeomView has most of the other features of Salem (in some ways GeomView might be considered to be a successor to Salem). G... |

36 | Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology, Par t 2 , Stability considerations - Moffatt - 1986 |

35 |
A Topological Picturebook
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(Show Context)
Citation Context ...8]. Several books have been published in the past decade that are superb examples of mathematical visualization. Perhaps the best of these is A Topological Picturebook, George Francis’ beautiful book =-=[Fra87]-=- on many different aspects of topology (other work by Francis is equally impressive, for example [FC92]). Most of the illustrations are hand-drawn diagrams of 3D objects, but he does include a chapter... |

35 | Cluster busting in anchored graph drawing - Lyons - 1992 |

34 |
A calculus for rational tangles: Applications to DNA Recombination
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- 1990
(Show Context)
Citation Context ...e greatest single contribution of Conway’s paper is the concept of the tangle itself. The reader is urged to consult the original paper [Con70] or more recent work that focuses on the tangle calculus =-=[ES90]-=- or uses tangles in innovative ways [Rol94] to learn more about these interesting applications. A precise grammar for the Conway notation may be found in the KnotPlot manual [Sch97b]. 3.3 Topological ... |

31 |
The relative motion of the earth and the luminiferous ether
- Michelson
- 1881
(Show Context)
Citation Context ...th areas where topological drawing can be of use in physical knot theory. The following are a sample of some of the topics that will be of interest in later chapters. 13 Prior to Michelson and Morley =-=[MM87]-=- providing strong evidence for its non-existence the ether was generally believed to be an extremely tenuous fluid that permeated the universe and acted as the medium in which electromagnetic waves we... |

30 | Drawing graphs in the plane with high resolution - Formann, Hagerup, et al. - 1993 |

29 | Topological evolution of surfaces
- DeCarlo, Gallier
- 1996
(Show Context)
Citation Context ...hosis between two different objects). Although most researchers in the area are well aware of the difficulties of morphing between two arbitrary three-dimensional objects [Hug92], DeCarlo and Gallier =-=[DG96]-=- point out that many do not take topology seriously. The fact that all the power of modern mathematics is yet unable to effectively “handle” cases such as Figure 1.1 indicates that if we do take topol... |

27 |
On types of knotted curves
- Alexander, Briggs
- 1927
(Show Context)
Citation Context ...s, is given in Rolfsen’s book Knots and Links [Rol76]. This catalogue, arguably the most well-known of all knot catalogues, depends on two important earlier enumerations, that of Alexander and Briggs =-=[AB27]-=- and another by Conway [Con70]. Alexander and Briggs produced the first complete catalogue of knots up to nine crossings in which each knot was known to be distinct from others in the catalogue. Conwa... |

27 |
Classification of knot projections, Topology and its applications 16
- Dowker, Thistlethwaite
- 1983
(Show Context)
Citation Context ...ugh most knots (knot diagrams) have many DT-codes, one of these will be minimal in a lexicographical sense and that one may be chosen as the standard “name” for the knot (diagram). The enumeration in =-=[DT83]-=- follows this convention, however, none of the methods described in what follows are affected by which code is used. Granny knot Square knot in alternating projection Figure 6.3: Different composite k... |

24 | The Classi of Knots and 3{Dimensional Spaces - Hemion - 1992 |

22 | Visualizing quaternion rotation
- Hart, Grancis, et al.
- 1994
(Show Context)
Citation Context ...4D objects. As was discussed in Chapter 1, a visualization goes beyond mere display if it serves to elucidate the concept in mathematics that it is depicting. The paper by Hart, Francis, and Kauffman =-=[HFK94]-=- provides an effective visualization of quaternion algebra and its relationship to rotations in 3D. Particularly interesting is the illustration of the counter-intuitive “Dirac string trick” (also kno... |

22 |
theorems for knots, links and spatial graphs
- Ramsey
- 1991
(Show Context)
Citation Context ...is known: Table 3.2 contains most of the known values. • The connected sum of N trefoils has a stick number of 2N +4. • For a knot K with crossing number c(K), the stick number s(K) is constrained by =-=[Neg91]-=- 5+ � 8c(K)+9 ≤ s(K) ≤ 2c(K) 2 The lower bound is fairly weak, since this implies that knots with ten crossings may have stick numbers as low as eight. Experimental results ([Mei96] and Chapter 8) ind... |

22 |
A set of topological invariants for graphs
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- 1933
(Show Context)
Citation Context ... we now see the reason for the DT-code specifying a unique knot diagram. It is related to the fact that 3-connected planar graphs have a unique embedding, up to homeomorphisms of the (extended) plane =-=[Whi33]-=-. We now establish that the quad graphs are indeed 3-connected in the following theorem. Theorem 1 Let g be the DT-code of a prime knot K in minimal projection and let G be the quad-graph obtained fro... |