Bobbin lace is a fibre art form in which intricate and delicate patterns are created by braiding together many threads. We present an overview of the making of bobbin lace, illustrated with a simple, traditional bookmark design. We briefly summarize research on the topology of textiles and braid theory, which form a base for the current work. We then define a new mathematical model that supports the enumeration and generation of bobbin lace patterns using an intelligent combinatorial search, and which is capable of producing patterns that have never been seen before. Finally, we apply our new patterns to an original bookmark design and propose future areas for exploration.
An is a braid in which each strand alternates going over and under the strands that it crosses (see Figures c and d). Alternating braids are characterized by the property that the @math generators for even positions have the opposite sign (superscript) from the @math generators for odd positions @cite_1 . Given the generator representation for bobbin lace actions, we infer that any combination of cross and twist will result in an alternating braid.
@cite_1
2137459590 2410271282 1616071251 2168485256
IN THIS PAPER I construct a state model for the (original) Jones polynomial [5]. (In [6] a state model was constructed for the Conway polynomial.) As we shall see, this model for the Jones polynomial arises as a normalization of a regular isotopy invariant of unoriented knots and links, called here the bracket polynomial, and denoted 〈K〉 for a link projectionK . The concept of regular isotopy will be explained below. The bracket polynomial has a very simple state model. In §2 (Theorem 2.10) I use the bracket polynomial to prove (via Proposition 2.9 and an observation of Kunio Murasugi) that the number of crossings in a connected, reduced alternating projection of a link L is a topological invariant of L. (A projection is reduced if it has no isthmus in the sense of Fig. 5.) In other words, any two connected, reduced alternating projections of the link L have the same number of crossings. This is a remarkable application of our technique. It solves affirmatively a conjecture going back to the knot tabulations of Tait, Kirkman and Little over a century ago (see [6], [9], [10]). Along with this application to alternating links, we also use the bracket polynomial to obtain a necessary condition for an alternating reduced link diagram to be ambient isotopic to its mirror image (Theorem 3.1). One consequence of this theorem is that a reduced alternating diagram with twist number greater than or equal to one-third the number of crossings is necessarily chiral. The paper is organized as follows. In §2 the bracket polynomial is developed, and its relationship with the Jones polynomial is explained. This provides a self-contained introduction to the Jones polynomial and to our techniques. The last part of §2 contains the applications to alternating knots, and to bounds on the minimal and maximal degrees of the polynomial. §3 contains the results about chirality of alternating knots. §4 discusses the structure of our state model in the case of braids. Here the states have an algebraic structure related to Jones’s representation of the braid group into a Von Neumann Algebra. The theory of braids shows the interplay of two disciplines of pure mathematics? topology, used in the definition of braids, and the theory of groups, used in their treatment. The fundamentals of the theory can be understood without too much technical knowledge. It originated from a much older problem in pure mathematics?the classification of knots. Much progress has been achieved in this field ; but all the prog ress seems only to emphasize the extreme difficulty of the problem. Today we are still very far from a complete solution. In view of this fact it is advisable to study objects that are in some fashion similar to knots, yet simple enough so as to make a complete classification possible. Braids are such objects. In order to develop the theory of braids we first explain what we call a weaving pat tern of order ( being an ordinary inte gral number which is taken to be 5 in Fig. 1). Let Li and L2 be two parallel straight lines in space with given orientation in the same sense (indicated by arrows). If is a point on Lh Q a point on L2, we shall sometimes join and Q by a curve c. In our drawings we can only indicate the pro jection of c onto the plane containing L It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model[13, 3, 1]. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maximally mixed state. One clean qubit computers are believed to be strictly weaker than standard quantum computers, but still capable of solving some classically intractable problems [21]. Here we show that evaluating a certain approximation to the Jones polynomial at a fifth root of unity for the trace closure of a braid is a complete problem for the one clean qubit complexity class. That is, a one clean qubit computer can approximate these Jones polynomials in time polynomial in both the number of strands and number of crossings, and the problem of simulating a one clean qubit computer is reducible to approximating the Jones polynomial of the trace closure of a braid. In this paper we study the problem of hair interpolation: given two 3D hair models, we want to generate a sequence of intermediate hair models that transform from one input to another both smoothly and aesthetically pleasing. We propose an automatic method that efficiently calculates a many-to-many strand correspondence between two or more given hair models, taking into account the multi-scale clustering structure of hair. Experiments demonstrate that hair interpolation can be used for producing more vivid portrait morphing effects and enabling a novel example-based hair styling methodology, where a user can interactively create new hairstyles by continuously exploring a “style space” spanning multiple input hair models.