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• W80768568 abstract "A recursive formula has been discussed, which involves the Arf invariant, the special value of the Alexander polynomial and the Jones polynomial, and the Minkowski unit of a knot. Also, we reformulate J. Levine’s result on the Kervaire invariant of a homotopy sphere as a recursive formula, which suggests that the smooth structure of spheres would be distinguished from number theoretical reasons. 1 Arf invariant and Legendre symbol Let p and q be different odd primes such that p, q ≡ 1 (mod 4). Then the mod 2 linking number, lk2(p, q) ∈ Z2, of p and q is defined by the equality ( q p ) = (−1)2, (1) where the Legendre symbol (q/p) is defined to be either +1 or −1 according as q is or is not a quadratic residue modulo p. Note that the symmetry of the mod 2 linking number, lk2(p, q) = lk2(q, p), is nothing but the Gauss reciprocity law. However, the equality (1) is originally not a definition but a theorem. In fact, lk2(p, q) is defined as the image of the Frobenius automorphism σp ∈ π1(Xq) over p in Gal(Yq/Xq) ∼= Z2, where Yq → Xq := Spec(Z) − {q} is the double Etale covering. The mod 2 linking number is not always defined for any primes. For example, lk2(p, 2) is ill-defined. As is indicated in [8] by Morishita, there are many analogies between knots and primes by nature. The purpose of this note is to observe various knot invariants from number theoretical viewpoint and further discuss its higher dimensional analogue. A Note On A Recursive Formula Of The Arf-Kervaire Invariant 67 In [9] Murakami proved a recursive formula (see [4, Theorem 10.6] also) as follows: VL(i) = (− √ 2)r−1(−1)Arf(L) for a proper r-component link L, where Arf(L) denotes the Arf invariant of a link L, VL(t) is the Jones polynomial of L (see [4] for the definitions) and i = √ −1. In particular, if L is a knot, then we have VL(i) = (−1). (2) Given a link, the special value of the polynomial invariant often play an important role in studying links. It is well-known (see [4, Chapter 10] again) that for a knot K, Arf(K) = (∆K(−1)) − 1 8 (mod 2), (3) where ∆K(t) is the Alexander polynomial of a knot K and note that ∆K(−1) is odd for any knot K. Moreover, it is easy to see that VK(−1) = ∆K(−1) by checking the skein relations, so we also have Arf(K) = (VK(−1)) − 1 8 (mod 2). (4) On one hand in number theory, the 2-nd reciprocity law due to Gauss tells us ( 2 p ) = (−1) p2−1 8 . (5) Thus we can reformulate (3) and (4) by applying (5) as a recursive formula: ( 2 ∆K(−1) ) = ( 2 VK(−1) ) = (−1). (6) In addition, Murasugi defined the Minkowski unit Cp(K) of a link for a prime p (see [10]), and he proved that C2(K) = (−1). (7) By (2) and (7) we immediately have VK(i) = C2(K). Hence by combining these formulas on knot invariants we have ( 2 ∆K(−1) ) = ( 2 VK(−1) ) = VK(i) = C2(K) = (−1). (8) Remark 1.1. For a knot K, ∆K(1) = ±1 and ∆K(−1) possibly takes any odd integer since the Alexander polynomial is symmetric in Z[t, t−1]. The Legendre symbol is defined only for primes but it is extended as the Jacobi symbol (q/p) with same notation, which is defined for any odd p. Note that the 2-nd reciprocity formula with the Jacobi symbol holds in the same formula as (5). Thus it should be regarded that (8) is formulated by using the Jacobi symbol." @default.
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• W80768568 date "2008-01-01" @default.
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• W80768568 title "A Note On A Recursive Formula Of The Arf-Kervaire Invariant" @default.
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