By Seiichi Kamada

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Let D be a maximal disk of F in the level at t = d . Consider an arc α in R4 connecting an interior point q of D and a point q in R3 × {d} such that α ∩ F = {q} and α ∩ R3 × {t} is a point for each t ∈ [d , d]. ) Let F1 be a surface in R3 × [a, ∞) obtained this way. Then F ∩ R3 × [a, c] = F1 ∩ R3 × [a, c], and F1 has no critical points or critical disks in R3 × (c, d), and all maximal dasks are in R3 × {d}. 5, F1 is ambient isotopic to a surface F2 such that F2 ∩ R3 × [a, c] = F ∩ R3 × [a, c], F2 ∩ R 3 × (c, d) = L × (c, d) for a link L, and all maximal disks are in R3 × {d}.

Rolfsen [150]. A simple loop C on T is said to be of type-(p,q) if [C] = p[m] + q[l] = 0 ∈ H1 (T ). A knot is called a torus knot of type-( p, q) if it is equivalent to a knot on T that is of type-( p, q) (Fig. 33). A link L with μ components is called a torus link of type-(a, b) if a/μ and b/μ are co-prime integers and L is equivalent to a link on T each of whose components is of type-(a/μ, b/μ). A torus knot or a torus link of type-(a, b) is denoted by T (a, b) in this book. length n and the sequence (a1 , a2 , .

The HOMFLY-PT polynomial28 PL ( , m) is a link invariant that takes values in Z[ , −1 , m, m −1 ] and satisfies PO ( , m) = 1, PL + ( , m) + −1 PL − ( , m) + m PL 0 ( , m) = 0. H. Conway [29], where it was called the potential function and the relathion between the Alexander polynomial and the potential funtion was given there. R. Jones [60, 61]. H. Kauffman [87, 88] introduced a state model for the Jones polynomial. 28 HOMFLY is the initials of the authors, P. Freyd, D. Yetter, J. R. Lickorish, K.