Examples
For the standard embedded torus, the skew-symmetric form is given by (with respect to the standard symplectic basis), and the skew-quadratic refinement is given by with respect to this basis: : the basis curves don't self-link; and : a (1,1) self-links, as in the Hopf fibration. This form thus has Arf invariant 0 (most of its elements have norm 0; it has isotropy index 1), and thus the standard embedded torus has Kervaire invariant 0.
Read more about this topic: Kervaire Invariant
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