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By K. Ribet

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L²-invariants: theory and applications to geometry and K-theory

In algebraic topology a few classical invariants - resembling Betti numbers and Reidemeister torsion - are outlined for compact areas and finite crew activities. they are often generalized utilizing von Neumann algebras and their lines, and utilized additionally to non-compact areas and limitless teams. those new L2-invariants include very fascinating and novel details and will be utilized to difficulties bobbing up in topology, K-Theory, differential geometry, non-commutative geometry and spectral thought.

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Since f is an invariant of C3 -equivalence and a2 is additive under the connected sum of knots, we have f (L) = f (L ). The following is the main result in this section. 1. Let L and L be 2-string links. Then L is self C2 -equivalent (self delta-equivalent) to L if and only if µL (12) = µL (12), µL (1122) = µL (1122), and f (L) = f (L ). March 4, 2007 11:41 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 32  L B! Fig. 5. A local C2 -clasper. 1 and 4. 3. Let T1 (resp. T2 ) be a simple Ck -clasper (resp.

Providence, RI, 2006. 5. H. Goda and A. Pajitnov, Twisted Novikov homology and Circle-valued Morse theory for knots and links, Osaka J. Math. 42 (2005), 557–572. 6. M. Hirasawa and L. Rudolph, Constructions of Morse maps for knots and links, and upper bounds on the Morse-Novikov number, preprint. 7. T. Kanenobu, The augmentation subgroup of a pretzel link, Math. Sem. Notes Kobe Univ. 7 (1979), 363–384. 8. J. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. ; University of Tokyo Press, Tokyo 1968.

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. Figure 4 (a) depicts a Bennequin surface for K, obtained by our algorithm. Braiding other minimal genus Seifert surfaces for K, we obtain other minimal braids (Figure 4, (b) and (c)). O. Kakimizu [6] classified incompressible Seifert surfaces for prime knots of ≤ 10 crossings up to isotopies respecting the knot.

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