Casson Invariant - Properties

Properties

  • If K is the trefoil then .
  • The Casson invariant is 2 (or − 2) for the PoincarĂ© homology sphere.
  • The Casson invariant changes sign if the orientation of M is reversed.
  • The Rokhlin invariant of M is equal to half of the Casson invariant mod 2.
  • The Casson invariant is additive with respect to connected summing of homology 3-spheres.
  • The Casson invariant is a sort of Euler characteristic for Floer homology.
  • For any let be the result of Dehn surgery on M along K. Then the Casson invariant of minus the Casson invariant of

is the Arf invariant of .

  • The Casson invariant is the degree 1 part of the LMO invariant.
  • The Casson invariant for the Seifert manifold is given by the formula:

\lambda(\Sigma(p,q,r))=-\frac{1}{8}\left[1-\frac{1}{3pqr}\left(1-p^2q^2r^2+p^2q^2+q^2r^2+p^2r^2\right) -d(p,qr)-d(q,pr)-d(r,pq)\right] where d(a,b)=-\frac{1}{a}\sum_{k=1}^{a-1}\cot\left(\frac{\pi k}{a}\right)\cot\left(\frac{\pi bk}{a}\right)

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