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Math. Helv. 61 (1986), 519555. 2. H. Goda, Heegaard splitting for sutured manifolds and Murasugi sum, Osaka J. Math. 29 (1992), 21-40. 3. H. Goda, On handle number of Seifert surfaces in S 3 , Osaka J. Math. 30 (1993), 63-80. 4. H. Goda, Circle valued Morse theory for knots and links, Floer Homology, Gauge Theory, and Low-Dimensional Topology, 71-99, Clay Math. , 5, Amer. Math. , Providence, RI, 2006. 5. H. Goda and A. Pajitnov, Twisted Novikov homology and Circle-valued Morse theory for knots and links, Osaka J.

Bring the unshaded disk to the bottom of the shaded disk making the horizontal bands vertical (Figure 3, right). Now, we have braided the Seifert surface, and since there are B(L) + j (|Cj | − 1) + 1 = b(L) horizontal disks, the boundary of this surface is a minimal string braid for L. Fig. 3. Braiding a 2-bridge link surface 5. Some examples Let K be the knot S(41, 24), also known as 918 . Since 24/41 = [2, 4, 2, 4], b(K) = (0 + 1 + 0 + 1) + 1 + 1 = 4, and g(K) = 2. So the minimalstring, minimal-length braid has four strings and seven bands.

S. Sikora Categorification of the Kauffman bracket skein module of I−bundles over surfaces. Algebraic and Geometric Topology 4, 2004, 1177-1210. 2. G. E Bredon, Introduction to compact transformation groups. Acdemic Press (1972) 3. V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. ) 12 (1985), no. 1, 103–111. 4. L. H. Kauffman, An invariant of regular isotopy. Trans. Amer. Math. Soc. 318 (1990), no. 2, 417–471. 5. M. Khovanov, Categorification of the Jones polynomial.

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(83, 64)-Konfigurationen in Laguerre-, Mobius-und weiteren Geometrien by Benz W.

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