Chicago, Illinois and Galais, France November 15, 2000 xv Table of Contents Preface t o the First Edition. And finally, throughout there is a deep underlying movement in the relation between space and sign, geometry and symbol, the source common to mathematics, physics, language and all. Nevertheless, it is interesting to design such a model directly on the oriented diagrams. This first part of the book is a short course on knots and physics. Show that K where the integral denotes a Berezin integral. . To return to the crossed lines, they are not quite innocuous.
With the introduction of weaving - as in the clove hitch - interlocking tensions and frictions can produce an excellent bind. In order to do this it is helpful to first give the form of Wittens definition and then play with this form in an elementary way. These 3-manifold invariants were foretold by Edward Witten in his landmark paper on quantum field theory and the Jones polynomial. Each state is weighted by the contributions of its decorations. This article provides an introduction to the relationships between Vassiliev invariants, Knotsevich integrals and Wittens approach to link invariants via quantum field theory.
. Let K be any oriented link diagram. Alexander proved this result by regarding a closed braid as a looping of the knot around Thinking of three-dimensional space as a union of half-planes, each sharing the axis, we require that 6 intersect each half-plane in the same number of points - the number of braid-strands. In this case the hand-choice reduces the calculation by an enormous amount. I leave it as an exercise for the reader to see that normalization to an ambient isotopy invariant yields the usual version of the Jones polynomial. Knot Theory and Physics Louis H. As in the second edition, there is a selection of papers by the author at the end of the book.
The equivalence relation on diagrams that is generated by all four Reidemeis- ter moves is called ambient isotopy. Since L and L' are ambient isotopic, it follows that b and ti are also ambient isotopic. . Hence ti can be obtained from 6 by a sequence of Markov moves of the type l. . That is, X is the sum of the linking number of K1 with the remaining components of K. This will have to wait for the next time! This is the same as saying that P and 7 P are inverse matrices.
Thus, if we label the lines of T K with indices, then T K corresponds to a formal product - or rather a sum of products, since repeated indices connote summations: If there is a commutative ring R and an index set 1 such that a , b, c,. Multiplicatively, V, is generated by the elements 1, e l , ez,. Thus we have proved the Theorem 8. There are, however, a few other cases of the type I11 move. Crane, 2-d physics and 3-d topology, Comm.
The specific form of the averaging is as follows: Let u be a state of K. We can immediately translate the conditions of these relativized Reidemeister moves to a set of conditions on the abstract tensors for creation, annihilation and interaction that will guarantee that T K is an invariant of regular isotopy. In this new edition, an article on Virtual Knot Theory and Khovanov Homology has beed added. Thus the square knot is depicted as the closed loop shown in Figure 5. The simplest tangle replacements for a 4-valent vertex are the two crossings, positive and negative, and the oriented smoothing. The piecewise linear approach using a rapidity parameter 0 shows us the pattern of this tensor in a new light. This angular correspondence is remarkable, and it beckons for a deeper physical interpretation.
There may also be a few residual Kronecker deltas curves with no critical points vis-a-vis this height function : t Example. This stance has the advantage of providing direct access to the algebra and to the combinatorial topology, as well as physical ideas. The non-commutativity for elements of P is essential. This loop is secure under tension at A. This paper shows how the Kontsevich integral approach to the Vassiliev invariants is directly related to the perturbative expansion of Witten's functional integral. With this convention, and assuming that A has been given a value in C, we see that Vk and V2ut are finitely generated over C.
Let's concentrate on the case We want to show that these two choices of template will give identical calculations in the state summations. I will consider only the case n odd, and verify two key cases. Let C t K denote the light crystal corresponding to a given knot or link K. Thus in Figure 7, we have shown the trefoil, the Hopf link, the Borromean rings and their shadows. Since L and L' are ambient isotopic, it follows that b and ti are also ambient isotopic. For c E C n, r I shall abbreviate A; for Xc l Xc 2.
The other case is left for the reader. The Rubber Band and Twisted Tube -- 3. This stance has the advantage of providing direct access to the algebra and to the combinatorial topology, as well as physical ideas. We shall do this and use the notation V: and':V for these quotient mod- ules. Sections 12° and 13° discuss spin networks and q-deformed spin networks. In taking over the Yang-Baxter equation for topological purposes, we can use the same intepretation, but think of the diagrams with their under- and over-crossings as modeling events in a spacetime with two dimensions of space and one dimension of time. E 102 We need to check certain things, and some comments are in order.