On knot-spanning surfaces: an illustrated essay on topological art: with an artist's statement by Brent Collins
George K. Francis, Brent Collins · 1993
shapes from single blocks of cedar wood. Many of his pieces are invariant under discrete spatial symmetry groups, and all of them involve continuously varying sectional curvatures. As a result, the precision and fidelity with which corresponding parts match, together with the subtle permutation of elementary constituent forms, are most attractive to mathematically educated viewers. Collins is not a mathematician. The sculpture shown in Fig. 1 is representative of one of Collins's series of motifs. Apart from the base, it is clearly invariant under 180? rotations about all three principal axes-in a kind of three-dimensional (3D) playing-card symmetry. (For another example of this kind of symmetry, see Color Plate A No. 2.) Such symmetries are very helpful for inferring details of shape from only one picture. Naturally, it would be much more satisfactory to view the piece from all angles or, even better, to view it while rotating it manually. The object is of uniform 1/4-in thickness, and so may be regarded as a material realization of an abstract surface whose mathematical nature I shall explore presently. This abstraction does immediate violence to two further features of mathematical interest, the wood grain and the edge of the sculpture. The former has more aesthetic than technical value. The wood grain frequently suggests the illustrations one finds in treatises on Morse Theory [4]. When the concentric cylinders of dicotyledonous wood are cut transversely by a plane, they form the familiar tree rings. Cutting by domeshaped or saddle-shaped surfaces brings out the characteristic markings of a Morse function in the vicinity of its singularities. 314 Francis with (ollins, On Knot-Spanning Surfaces This content downloaded from 157.55.39.104 on Sun, 19 Jun 2016 06:38:07 UTC All use subject to http://about.jstor.org/terms The second feature has a technically highly interesting aspect that we shall ignore after this paragraph. From the sharp, uniformly /4-in-wide edges of the artwork, we abstract closed, knotted and linked ribbons curving through space. The mathematical surface depicted by the artwork may thus be regarded as a spanning surface of aframed link. Rolfsen's Knots and Links is an authoritative treatment of this subject [5]. We shall ignore the question of how this ribbon is twisted along itself and consider the abstract surface ending along an unframed link (Fig. 2) as its border. Collins's surface (see Fig. 1) has two border curves that are not linked. The little one in the center, shaped like the numeral 8, plays a special role. Imagine that the artist had not yet penetrated the center panel, but left a smooth sheet. This way, the surface spans a single, knotted curve. If it is carefully traced, it can be seen to be shaped like the knot shown in Fig. 3a. (As it will become necessary to compare the shapes in Fig. 3, they appear together in the same figure.) Collins's surface shown in Fig. 1, without the central detail, corresponds the surface shown in Fig. 3b. I shall discuss the significance of this surface later. The knot spanned by Collins's surface shown in Fig. 1 goes by many names: figure-8 knot, four-knot and Listing's knot, whose namesake coined the term topology to describe the study of shapes and spatial transformations of abstract objects. This knot is a respectable character of an area of mathematics known, appropriately enough, as knot theory. Rolfsen's Knots and Links [6] is an ample portal to this fascinating kind of mathematics and its useful bibliography points to the remaining important references. However, a small subset of what there is to say about this knot can also be found in A Topological Picturebook [7]. A BRIEF TOPOLOGY LESSON Wood is a rigid substance, so much so that it is difficult to imagine a surface carved from wood as having the quality of stretchable rubber. It also seems wasteful to forget the sensuous curvature Collins carves into his sculptures. Yet these operations are necessary for studying an object in topology. Two objects are topologically equivalent, or homeomorphic, if one can be distorted so as to correspond point for point with the other. Actually, this notion is more correctly a common description of an isotopy. The notion of topological equivalence is more generous, allowing for a temporary dismemberment of the object as long as corresponding 'rips' are faithfully 'sewn' together again. Moreover, an isotopy must be realizable in its entirety in space. A homeomorphism may be described piecemeal and unpictorially. In deforming one topological object into an equivalent one, we do not allow knots to come apart, or surfaces to pass through each another. Within these tolerable restrictions, topologists have been able to solve the classification problem for knots and for the surfaces spanning them. In order to understand just what it means to say that there are only six different surfaces spanning Listing's knot, we need to establish a few more notions from the topology of surfaces [8]. A surface is said to be closedif it has no boundary (Fig. 4c). Some are one-sided [9], such as a M6bius band (Fig. 4b) or a Klein bottle (Fig. 4d). (In addition to Fig. 1, Collins's piece in Fig. 7 is one-sided; the rest of his sculptures shown here are two-sided. The border curves of Fig. 8 naturally extend to infinity. Thus this may be regarded as a part of an infinite surface with three ends. The border curves of Color Plate A No. 2 and Fig. 9 look like the rims of discs that have been removed from two closed surfaces of higher genus.) Compact surfaces, that is, finite surfaces that are either closed or bordered, are classified by three qualities: (1) by the number of component curves contained in their boundary, (2) by whether or not they are two-sided and (3) by an integer presently defined as the Euler characteristic. This means that any two compact surfaces are topologically equivalent or homeomorphicunless they differ in one or more of these qualities. This is the fundamental theorem of the theory of surfaces. To determine the orientability is a matter of tracing your mind's finger along the curve. The Euler characteristic is more difficult to define. I shall first give an operational definition. Two other definitions follow. The disc, naturally, shall have Euler characteristic 1 [10]. Sewing two surfaces together along the entirety of a border circle on each produces a surface whose Euler number shall be the sum of the constituents. Hence, a sphere has Euler characteristic 2 (for the two hemispheres sewn on the equator), and the Euler characteristic of a surface decreases by one for each disc removed. Consistent with this is the rule that cutting across a ribbon increases the Euler characteristic by 1. Thus, the characteristic of a Mobius band is 0 (one less than that of the disc) [11 ]. Two surfaces can always be connected in the following way to produce a third surface. Replace two discs, one on each surface, by a cylindrical tube that connects their border circles. The result of such an operation is called the connected sum of the constituent surfaces. The Euler characteristic of the connected sum of two surfaces is two less than the sum Fig. 4. (a) A torus, (b) a Mobius band, (c) another torus with a circular window removed and (d) a Klein bottle. The torus and the Klein bottle are closed surfaces; the Mobius band and holey torus have a single border curve. The Klein bottle and Mobius band are one-sided surfaces.