By Andrew J. Casson
This ebook, which grew out of Steven Bleiler's lecture notes from a direction given via Andrew Casson on the collage of Texas, is designed to function an advent to the purposes of hyperbolic geometry to low dimensional topology. particularly it offers a concise exposition of the paintings of Neilsen and Thurston at the automorphisms of surfaces. The reader calls for in basic terms an realizing of easy topology and linear algebra, whereas the early chapters on hyperbolic geometry and geometric buildings on surfaces can profitably be learn via a person with a data of ordinary Euclidean geometry meaning to research extra abour different 'geometric structures'.
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Additional info for Automorphisms of Surfaces after Nielsen and Thurston
Let K be a knot. Then a knot diagram of K consists of a sequence of crossings of two pieces of curves cut from the knot K where the ordering of the crossings can be determined by the orientation of the knot K. As an example we may consider the two trefoil knots in the above section. Each trefoil knot is represented by three crossings of two pieces of curves. These three crossings are ordered by the orientation of the trefoil knot starting at z1 . Let us denote these three crossings by 1, 2 and 3.
Let us set z1 = z4 . In this case the quantum Wilson line forms a closed loop. Now in (79) with z1 = z4 we have that e−t log ±(z1 −z2 ) and et log ±(z1 −z2 ) which come from the two-side KZ equations cancel each other and from the multivalued property of the log function we have W (z1 , z1 ) = Rn A14 n = 0, ±1, ±2, ... (90) where R = e−iπt is the monodromy of the KZ equation . Remark. It is clear that if we use other representation of the quantum Wilson loop W (z1 , z1 ) (such as the representation W (z1 , z1 ) = W (z1 , w1 )W (w1 , w2 )W (w2 , z1 )) then we will get the same result as (90).
We want to prove that 12 = 21. This will give the subcircling property. Since a closed loop is formed we have that each open end of 1 or of 2 is connected to a closed loop. In this case as the above cases we have that the products 12 is with the initial operator A being a 2-tensor since the open ends of 1 or 2 do not cause A to be a tensor with tensor degree more than 2 by their connection to the closed loop. Indeed, let z be an open end of 1 or 2. Then it is an end point of a quantum Wilson line W (z, z ) which is a part of 1 and 2 such that z is on the closed loop formed by 1 and 2.
Automorphisms of Surfaces after Nielsen and Thurston by Andrew J. Casson