The First Coefficient of the Conway Polynomial

A formula is given for the first coefficient of the Conway polynomial of a link in terms of its linking numbers. A graphical interpretation of this formula is also given. Introduction. Suppose that L is an oriented link of n components in 53. Associated to L is its Conway polynomial V¿(z), which must be of the form VL(z) = z-1[a0 + alz2+ ■■■+amz2m\. Let VL(z) = VL(z)/z"~1. In this paper we shall give a formula for a0 = VL(0) which depends only on the linking numbers of L. We will also give a graphical interpretation of this formula. It should be noted that the formula we give was previously shown to be true up to absolute value in [3]. The author wishes to thank Hitoshi Murakami for bringing Professor Hosakawa's paper to his attention. We shall assume a basic familiarity with the Conway polynomial and its properties. The reader is referred to [1, 2, 4, 5 and 6] for a more detailed exposition. The fact that VL(z) has the form described above can be found in [4 or 6], for example. 1. A formula for V¿(0). Suppose L [Kv K2,... ,Kn) is an oriented link in S3. Let ltj = lk(AT,, Kj) if i +j and define /„ = -jLu-ii+ihjDefine the linking matrix ££, or S^(L), as JSf= (/, ). Now JSPis a symmetric matrix with each row adding to zero. Under these conditions it follows that every cofactor =S?;y of £? is the same. (Recall thatSetj = (-l)'+ydet MtJ, where Mi} is the (i, j) minor of &.) Theorem 1. Let L he an oriented link of n components in S3. Then V¿(0) = 3?¡j, where ^j is any cofactor of the linking matrix Jif. Proof. Let F be a Seifert surface for L. We may picture F as shown in Figure 1.1. Let {a¡} be the set of generators for Ha[F) shown in the figure and define the Siefert matrix V = {v¡¡) in the usual way. Namely, vii} = lk(a,+, af), where a,+ is obtained by lifting a¡ slightly off of F in the positive direction. Then if a, n a, = 0 we have v¡, = v¡¡ = lk(a,, üj). If a¡ n Oj # 0, then {;', j) = {2k — 1,2/c) for some Received by the editors August 14, 1984 and, in revised form, December 28, 1984. 1980 Mathematics Subject Classification. Primary 57M25.