Lee Rudolph

The Fullness of Time

In this speculative article, I question the postulation of homogeneity in models of time intended for use in developmental, social or cultural psychology. After surveying extant models, I propose a new kind of model, which is highly inhomogeneous and one-dimensional only in a global sense: locally such a model can be very ‘full’. My aim is to capture in mathematical form one common hypothesis about psychological time (that it reckons ‘acts of attention’), somewhat amplified by a further claim (that these acts are interpolated by states of ambivalence), and to do so precisely enough that the resulting models can be used effectively in further collaborations between mathematicians and psychologists

Knot Theory of Complex Plane Curves

Publisher Summary The knot theory of complex plane curves draws attraction not only for its own internal results but also for its intriguing relationships and interesting contributions elsewhere in mathematics. Within low-dimensional topology, related subjects include braids, concordance, polynomial invariants, contact geometry, fibered links and open books, and Lefschetz pencils. Within low-dimensional algebraic and analytic geometry, related subjects include embeddings and injections of the complex line in the complex plane, line arrangements, Stein surfaces, and Hilberts 16th problem. There is experimental evidence that nature favors quasipositive knots. This chapter summarizes studies that relate the knot theory of complex plane curves to other research areas, highlighting the results from few studies. There is an enormous, interesting, and constantly growing literature on braid monodromy and its many offsprings, including applications to line arrangements and the Alexander invariants of complex plane curves.

The slice genus and the Thurston-Bennequin invariant of a knot

For any knot K ⊂ S3, gs(K) ≥ (TB(K) + 1)/2. Let K ⊂ S = ∂D ⊂ C ⊂ CP be a (smooth) knot. The slice genus gs(K) is the smallest genus of a smooth, oriented surface in D with boundary K. If K is Legendrian, that is, everywhere tangent to the field of complex lines tangent to S, then the Thurston-Bennequin invariant tb(K) is the integer associated in the usual way (using linking numbers in S, with the orientation induced by C) to the normal linefield to K which assigns to each point of K the tangent line to K multiplied by √−1; cf. [1]. For arbitrary K, the maximal Thurston-Bennequin invariant TB(K) is sup{tb(K ′) : K ′ ⊂ S is a Legendrian knot ambient isotopic to K}. Appealing to a theorem of Donaldson, I showed [4] that if gs(K) = 0, then TB(K) ≤ −1. The following inequality is more general; the proof requires more than Donaldson’s theorem, namely, the truth of the local Thom Conjecture [2]. Theorem. gs(K) ≥ (TB(K) + 1)/2. Proof. Using the construction of entire totally-tangential C-links given in [4], and elementary rational approximation theory, it is easily seen that, for any integer t ≤ TB(K), there is a rational immersion F : CP → CP, in general position, such that A = F−1(D4) ⊂ C ⊂ CP is a neighborhood of S diffeomorphic to an annulus, F |A is an embedding, and ∂F (A) = ∂A(K, t) ⊂ S, where ∂A(K, t) ⊂ S denotes an (embedded) annulus of type K with t twists (i.e., each component of ∂A(K, t) is ambient isotopic to K, and the linking number of the components is −t). Let Di ⊂ CP, i = 1, 2, be the two 2-disks complementary to IntA. Let the smoothly immersed 2-disk F (Di) ⊂ CP \ IntD have di doublepoints; let Si ⊂ CP be a smoothly immersed surface which is the union along a component of ∂A(K, t) of F (Di) and a suitable smoothly embedded surface in D 4 of genus gs(K). Let ni be the degree of Si. Then the degree of F (CP ) is n1 +n2, and, by the local Thom Conjecture, gs(K) + di ≥ (ni − 1)(ni − 2)/2. Let p be the intersection number of the relative cycles F (D1) and F (D2) in CP \ IntD. Then p− t = n1n2 is the intersection number of S1 and S2 in CP, and p+ d1 + d2 = (n1 + n2 − 1)(n1 + n2 − 2)/2. Received by the editors October 12, 1995. 1991 Mathematics Subject Classification. Primary 57M25; Secondary 14H99.

Stratified Deformation Space and Path Planning for a Planar Closed Chain with Revolute Joints

Given a linkage belonging to any of several broad classes (both planar and spatial), we have defined parameters adapted to a stratification of its deformation space (the quotient space of its configuration space by the group of rigid motions) making that space “practically piecewise convex”. This leads to great simplifications in motion planning for the linkage, because in our new parameters the loop closure constraints are exactly, not approximately, a set of linear inequalities. We illustrate the general construction in the case of planar nR loops (closed chains with revolute joints), where the deformation space (link collisions allowed) has one connected component or two, stratified by copies of a single convex polyhedron via proper boundary identification. In essence, our approach makes path planning for a planar nR loop essentially no more difficult than for an open chain.

A characterization of quasipositive Seifert surfaces (constructions of quasipositive knots and links, III)

Here, asurfaceis smooth, compact, oriented, and has no component with empt y boundary. A Seifert surfaceis a surface embedded in S3. A subsurfaceSof a surfaceT is full if each simple closed curve on S that bounds a disk on T already bounds a disk on S. The definition of quasipositivity is recalled in §1, after a review of braided surfaces. The “only if” statement of the Characterization Theorem is p roved in§2. Some results about graphs on braided surfaces (stated, with one eye on oth er applications [6], in somewhat more generality than needed here) are obtained in §3, and used in§4 to prove the “if” statement of the Characterization Theorem. A conjectural e xtension to ribbon surfaces in the 4-disk is discussed in §5.

A Unified Geometric Approach for Inverse Kinematics of a Spatial Chain with Spherical Joints

Conventionally, joint angles are used as parameters for a spatial chain with spherical joints, where they serve very well for the study of forward kinematics (FK). However, the inverse kinematics (IK) problem is very difficult to solve directly using these angular parameters, on which complex nonlinear loop closure constraints are imposed by required end effector configurations. In a recent paper, our newly developed anchored triangle parameters were presented and shown to be well suited for the study of IK problems in many broad classes of linkages. The focus of that paper was the parameterization of non-singular solutions; among many specific types of IK problems, only one, that of a spatial chain with spherical joints imposing 5 dimensional constraints, was developed in detail. Here we present a unified approach to the solutions of that and two other types of IK problems. The critical concepts in our approach - the geometric formulation in anchored triangle parameters, and the application of loop deformation spaces - are general for all IK problems, and especially useful for redundant systems. For the three IK problems addressed in this paper, we demonstrate convexity properties of the set of IK solutions. We also give detailed descriptions of the parameterization of singular deformations. Similar ideas apply readily to linkages involving multiple loops.

Spaces of ambivalence: Qualitative mathematics in the modelling of complex fluid phenomena

Abstract This paper sketches two classes of mathematical models. Both treat ambivalence and attentiveness as undefined terms. The first class, ambivalence-generated models, is finitistic and requires no ontological commitment to any mathematical construction as sophisticated as ‘real numbers’. The intended semantics suggests the axiom underlying topologists’ well-understood theory of finite simplicial complexes (FSCs). Semantically reinterpreted within psychology, this theory yields concrete, empirically testable hypotheses about human behaviour, which in turn suggest further axiomatic restrictions on the models. The second class of models treats attentiveness as a Morse function on some differential manifold, and uses its gradient flow to construct a lower-dimensional spine. Both classes of models have potential to capture much of the flexibility and concreteness that are attractive in qualitative methodologies, while retaining (by application of mathematical analysis) the formality of quantitative methodologies.

Difference index of vectorfields and the enhanced Milnor number

THE MILNOR number p( .X) of a simple fibered link X = (S*” - ‘, K) is now usually defined as the rank of the middle homology of a fiber F for the link, but it was in fact first introduced by Milnor for links of singularities as a certain mapping degree; its equality with a Betti number of F was a theorem (Milnor [S]; he called it “multiplicity”). In [lo] it was observed that an invariant introduced in [12] for fibered links in S3 could be defined in any dimensions and its definition could be seen as a natural extension of Milnor’s definition of p(X). For this reason it was named the enhanced Milnor number. Suitably normalized it takes the form (U(X), n(X)), lying in Z 8 Z or 2@2/2 according as the ambient dimension is 3 or > 3. The invariant E.(Y)EZ or Z/2 is called the enhancement to the Milnor number. The enhanced Milnor number is defined as follows. Let X = (S*” - r, K) be a fibered link. We define a (2n - 2)-plane field in the stabilized tangent bundle TS2” - 1 0 W as follows: outside a tubular neighborhood N of K we use the tangent field to the fibration for X, along K we use (tangent field to K) @ R, and over the rest of N we interpolate as directly as possible between the above fields on aN and K. TS*” - 1 @ R has a trivialization coming from the embedding of S*” - l in R*“, so the above field defines a mapping A:S*” - ’ + G(2n - 2,2n) to the Grassman manifold of 2n - 2 planes in R*“. The homotopy class of this mapping in 7r2” _ 1 G(2n - 2,2n) is the enhanced Milnor number. As described in [lo], this homotopy group is isomorphic to Z @ Z for n = 2 and to Z @ Z/2 for n > 2, and the enhanced Milnor number has the form (( - l)“p(X), A(X)), where p(X) is the usual Milnor number. In fact the first summand is the image of 7~~” _ l(GC(n - 1, n)) in 7~~” _ 1 G(2n - 2,2n), so I vanishes if and only if the above field of real (2n - 2)-planes can be homotoped to a field of complex (n - 1 )-planes in @” 2 R*“. For simple links j.(X) is determined modulo 2 by the Seifert form L:

Hopf plumbing, arborescent Seifert surfaces, baskets, espaliers, and homogeneous braids

Abstract Four constructions of Seifert surfaces—Hopf and arborescent plumbing, basketry, and T -bandword handle decomposition—are described, and some interrelationships expounded, e.g., arborescent Seifert surfaces are baskets; Hopf-plumbed baskets are precisely homogeneous T -bandword surfaces.

Convexly Stratified Deformation Spaces and Efficient Path Planning for Planar Closed Chains with Revolute Joints

Systems involving loops have been especially challenging in the study of robotics, partly because of the requirement to maintain loop closure constraints, conventionally formulated as highly nonlinear equations in joint parameters. In this paper, we present our novel triangle-tree-based approach and parameters for planar closed chains with revolute joints. For such a loop, the loop closure constraints are exactly, not approximately, a set of linear inequalities in our new parameters. Further, our new parameters provide explicit parametrization of the system deformation space (configuration space modulo the group of rigid motions of the systems ambient space respecting system specifications) and endow it with a nice geometry. More precisely, the deformation space of a generic planar loop with n revolute joints consists of 2n-2 copies of one and the same convex polytope (which, when all of the link lengths are fixed, is bounded and of dimension n - 3), glued together into either one connected component or two (ignoring collision-free constraints), via proper boundary identification. Such a completely solved, stratified space of convex strata will have profound implications for these systems and lead to great simplifications in many kinematics related issues. For example, in essence, our approach makes path planning for planar loops with revolute joints no more difficult than for open chains. We also briefly point out the connection and extension of the work presented here to other systems such as spatial loops with spherical joints and systems involving multiple loops.