# The Arf Invariant and the Jones Polynomial

• W. B. Raymond Lickorish
Part of the Graduate Texts in Mathematics book series (GTM, volume 175)

## Abstract

The original Arf invariant was an invariant of certain quadratic forms on a vector space over a field of characteristic 2. This can be applied to a quadratic form, closely associated to the Seifert form, on the first homology with ℤ/2ℤ coefficients of a Seifert surface of an oriented link L. The result is a fairly classical link invariant A(L) ∈ ℤ/2ℤ called the Arf (or Robertello) invariant of L ([111], [114]). It must, however, be stated at once that for this theory to work—that is, for A(L) to be defined—L must satisfy the condition that the linking number of any component with the remainder of the link should be an even number. Before the discovery of the Jones polynomial, efforts to find a sensible generalisation of the Arf invariant to all links met with no success. The Jones polynomial V (L) is, of course, always defined. As will be shown in what follows, evaluating V(L) when t = i (with t 1/2 = e iπ/4)gives
$$V{(L)_{(t = 1)}} = \{ _0^{{{( - \sqrt 2 )}^{\# L - 1}}{{( - 1)}^{A(l)}}}$$
where #L is the number of components of L. In a sense, this shows why a definition of A(L) for any link could not be found. Interpreted from the point of view of the Jones polynomial, this result gives one of the very few evaluations of the polynomial in terms of previously known invariants that can be calculated in “polynomial time” (see Chapter 16). This chapter will first explore the Arf invariant for vector spaces over ℤ/2ℤ and then effect liaison with the Jones polynomial.

## Keywords

Quadratic Form Simple Closed Curve Jones Polynomial Symplectic Base Alexander Polynomial
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