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- W1981844088 abstract "Let L be a three-component link all of whose linking numbers are zero. Write the Alexander polynomial of L as A(x, y, z) = (1 x)(l y)(l z)f(x,y, z). Then the integer If(l,l,l)I is a perfect square. In 1953, Torres [7] gave necessary conditions for a polynomial to be the Alexander polynomial of a link. These conditions have never been proved sufficient, even though the question has appeared on at least two important lists of knot and link theory problems ([3, p. 168, no. 2], [2, p. 218, no. 15]). In this paper, we give a new condition for a polynomial to be the Alexander polynomial of a three-component link, all of whose linking numbers are zero. According to the Torres conditions, the Alexander polynomial of such a link can be written A(x, y, z) = (1 x)(1 -y)(l Z)f(x, y, z). We prove that the integer If(1, 1, 1)I is a perfect square, and then give a class of examples to show that If(1, 1, 1)I can be any perfect square. Unfortunately, there are also three-component links with one nonzero linking number whose Alexander polynomials are divisible by (1 x)(I y)(l z), and in which If(1,j,j)I is not a perfect square. Thus we have not shown that the Torres conditions are insufficient in general, but only in the presence of a restriction on the linking numbers. The author would like to thank William Massey for his generous contribution to the discovery and proof of this theorem, and to thank the referee for raising an interesting question. 1. The main theorem. Seifert [6] gives an algorithm for spanning an orientable surface in any knot. He then represents this surface as a disk with 2h bands attached, where h is the genus of the surface. From this representation, he constructs the Seifert matrix, whose determinant is the Alexander polynomial. Hosokawa [4] (as well as Torres [7]) extends this procedure to links and their reduced Alexander polynomials by adding It 1 new bands to the above-mentioned disk and It 1 new rows and columns to the corresponding Received by the editors June 6, 1977 and, in revised form, October 5, 1977. AMS (MOS) subject classifications (1970). Primary 55A25." @default.
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- W1981844088 date "1978-02-01" @default.
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- W1981844088 title "On the Alexander polynomials of certain three-component links" @default.
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- W1981844088 doi "https://doi.org/10.1090/s0002-9939-1978-0482737-x" @default.
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