New developments in the theory of knots by Toshitake Kohno

By Toshitake Kohno

This reprint quantity specializes in contemporary advancements in knot concept coming up from mathematical physics, in particular solvable lattice versions, Yang-Baxter equation, quantum team and dimensional conformal box concept. This quantity is useful to topologists and mathematical physicists simply because latest articles are scattered in journals of many alternative domain names together with arithmetic and Physics. This quantity will provide an exceptional viewpoint on those new advancements in Topology encouraged through mathematical physics.

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Extra resources for New developments in the theory of knots

Example text

F : h −→ h f, f −1 holomorph . : offene Teilmenge von C), dann heißen X und Y biholomorph ¨ aquivalent, falls ein f : X −→ Y existiert, sodass f bijektiv ist und f, f −1 holomorph sind. 9 hist biholomorph ¨ aquivalent zu D := z ∈ C |z| < 1 . Beweis: Betrachte die Abbildung f : h −→ D, f (z) := z−i z+i . |f (z)| < 1 ⇐⇒ |z + i|2 > |z − i|2 ⇐⇒ (y + i)2 > (y − i)2 ⇐⇒ 2y > −2y ⇐⇒ y > 0 Bemerkung: 1. Die Umkehrabbildung zu f ist f −1 = f f −1 (w) = 1 w+1 i w−i 2. Es gibt zu f : h Aut(D). ): fA (z) = = az + b = cz + d (az + b)(c¯ z + d) |cz + d|2 (ad − bc) z |cz + d|2 Also fA ∈ Aut(h) ⇐⇒ A ∈ GL(2, R)+ .

1 i (Eigenwertgleichung). Die andere Richtung folgt durch Nachrechnen! 33 3 Die Riemannschen Fl¨achen C, C und h Also zu zeigen: Aut(D)0 = fN N = α 0 0 α = g ∃ s ∈ S 1 : g(z) = sz (Dabei ist S 1 := z ∈ C |z| < 1 ). Das Unterstrichene folgt aber aus dem folgenden Lemma. 2 Ist h ∈ Aut(D), dann |h(z)| ≤ |z|, dto. h. 11 (Lemma von Schwarz) Sei f : D −→ C holomorph, f (0) = 0, |f (0)| < 1 f¨ ur |z| < 1, dann gilt: 1. |f (z)| ≤ |z| ∀ |z| ≤ 1 2. Ist |f (z0 )| = |z0 | f¨ ur ein z0 = 0, dann ist f (z) = λz mit geeignetem λ ∈ C, |λ| = 1 1 Beweis: Setze g(z) := f (z) ur z da f (z) = 0, folgt g holomorph in D.

Damit ist ℘(z + γ) = ℘(z) + c(γ) (∗) (wobei c(γ) eine Konstante ist, die von γ abh¨ angt). Zeige c(γ) = 0: Offensichtlich ist ℘(−z) = ℘(z) (℘ ist eine gerade Funktion). 1 Ell(Γ) := f ∈ M er(C) ∀ γ ∈ Γ : f (z + γ) = f (z) heißt Ko ¨rper der elliptischen Funktionenelliptische Funktionen zu Γ. Konvention: Ist z0 Pol von f , dann setze f (z0 ) = ∞ ∈ C. 1 Ell(Γ) ist ein K¨orper. 1 Sei f ∈ Ell(Γ). F¨ ur z0 ∈ C, γ ∈ Γ gilt: ordz0 (f ) = ordz0 +γ (f ) Beweis: Es gibt eine offene Umgebung U von z0 und eine auf U holomorphe Funktion g mit g(z0 ) = 0, so dass f (z) = (z − z0 )n g(z) f¨ ur z ∈ U, z = z0 und n = ordz0 f (∗) (∗) Dann gilt aber f¨ ur z ∈ γ + U (γ + U ist offene Umgebung von z0 + γ): f (z) = f (z − γ) = n (z −(z0 +γ)) g(z −γ).

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