By Roger A. Johnson
This vintage textual content explores the geometry of the triangle and the circle, focusing on extensions of Euclidean conception, and reading intimately many quite fresh theorems. a number of hundred theorems and corollaries are formulated and proved thoroughly; quite a few others stay unproved, for use through scholars as routines. 1929 variation.
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Extra resources for Advanced Euclidean Geometry
Remark. The bracket state summation is an analog of a partition function [BA1] in discrete statistical mechanics . In fact, for appropriate choices of A, B, d the bracket can be used to express the partition function for the Potts model. See Part II sections 7° and 8° for this connection. 29 Example. From Figure 9 we see that the bracket polynomial for the trefoil diagram is given by the formula: (K) = A3d2-' + A2Bd'-' + A2Bd-' + AB2d2-1 + A2Bd1-' + AB2d2-1 AB2d2 -1 + B3d3-1 (K) = Aid' + 3A2Bds + 3AB2d1 + B3d2.
Thus CK has the potential to detect chirality. In fact, this is the case with the trefoil knot as we shall see. (i) (^ I = A(3 j + A-1 C^ > = A(-A3) + A 1(-A 3) (L) = -A 4 -A - 4 . '. '. -. ,CT. = -12 - A-16 A4 + A12 - A16. Since GT• # GT, we conclude that the trefoil is not ambient isotopic to its mirror image. Incidentally, we have also shown that the trefoil is knotted and that the link L is linked. (The Figure Eight Knot) 36 (E) = A(QR = A(-A ) +A-' ^Cj l ) ((& ) +A-' ^CG ) =-A 4 [-A 4 - A-4] + A-'[-A5 - A-3 +A -7] =A8+1-A4-A-4+A-8 (E)=A8 -A4+1-A-4+A-8.
Thus the normalized bracket yields the 1-variable Jones polynomial. 50 Remark. 1. Thus this theorem proves the existence (and well-definedness) of the 1-variable Jones polynomial. For the reader already familiar with VK(t) as satisfying (i), (ii), and (iii) the theorem draws the connection of this polynomial with the bracket. 2. )= B/-)+A(:) G) Hence (A _ B)( B-1 Thus A(<) - A-1( (A2 - A-2)(^ ). r )=w+landw (>4 )=w-1. Let a = -A3. 4 Letting A = t-1'4, we conclude that t-'5- tG = \'ft - f)G^W ^'!
Advanced Euclidean Geometry by Roger A. Johnson