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1. March 4, 2007 11:41 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 33 + 6 Fig. 7. 6 61 60 6: AS and IHX relations, where ⊕ in TX means a positive half twist. 1. 1. We will show the ‘if’ part. Let Ta , Tb , Tb , Tc , Tc and Td be claspers as illustrated in Figure 8, and let T x (x = a, b, c, d) be claspers obtained from Tx by adding a half-twist on an edge. Recall that string links have a natural monoid structure arising from composition. Let L be a 2-string link. 4, L is C2 -equivalent to the product of µL (12) copies A if µL (12) ≥ 0 or −µL (12) copies A−1 if µL (12) < 0, where A is a string link obtained from the trivial string link 12 by surgery along Ta , and A−1 is obtained from 12 by surgery along T a .

Math. de France, Ast´erisque 107-108 (1983) 87–161. 2. J. Birman and W. Menasco, Studying knots via braids III: Classifying knots which are closed 3 braids, Paciﬁc J. Math. 161 (1993) 25–113.

1, we have MN (L) = 2 × h(R) = 2. Note that L has a Seifert surface R such that h(R) = 2. March 4, 2007 11:41 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 40 Fig. 4. α1 α2 Fig. 5. 4. Connected sum The behavior of the handle number under a Murasugi sum has been studied in [2] and [3]. 1. Let K1 K2 be a connected sum of K1 and K2 . Then, MN (K1 K2 ) ≤ MN (K1 ) + MN (K2 ). However the study of the behavior of the Morse-Novikov number under a connected sum is not complete, that is, the next natural question is still open as far as I know.