A Simple Formula for the Casson–walker Invariant

Abstract

Gauss diagram formulas are extensively used to study Vassiliev link invariants. Now we apply this approach to invariants of 3-manifolds, considering a manifold as a result M L of surgery on a framed link L in S 3 . We study the lowest degree case -the celebrated Casson-Walker invariant λw of rational homology spheres. This paper is dedicated to a detailed treatment of 2-component links; a general case will be considered in a forthcoming paper. We present simple Gauss diagram formulas for λw(M L ). This enables us to understand/separate the dependence of λw(M L ) on L (considered as an unframed link) and on the framings. We also obtain skein relations for λw(M L ) under crossing changes of L, and study an asymptotic behavior of λw(M L ) when framings tend to infinity. Finally, we present results of extensive computer experiment on calculation of λw(M L ).

Universal invariants, the Conway polynomial and the Casson-Walker-Lescop invariant

“…Then S 3L is a rational homology sphere and λ L (M ) = det L|λ W (M ). One can easily check that (−1) σ−(L) is just the sign of det L, and Theorem 5.1 thus gives us1 2 det Lλ W (S 3 L ) = det L 8 σ(L) + µ 2 (L) − aµ 1 (K 2 ) − bµ 1 (K 1 ) .Using the explicit formulas for µ 1 and µ 2 given in Lemmas 4.5 and 4.6, we then obtain the following formula for det L 2 λ W (M ) − 1 4 σ(L) :ac 2 (L 2 ) + bc 2 (L 1 ) + n 3 − n 12 + (a + b) 24 (2n 2 − ab − 2) − c 3 (L) + n (c 2 (L 1 ) + c 2 (L 2 )) .This recovers a result of S. Matveev et M. Polyak[11, Thm. 6.3].The rest of this section is devoted to the proof of Theorem 5.1.…”

“…These two invariants are involved in the case n = 2 of Theorem 1, which recovers a theorem of S. Matveev and M. Polyak [11,Thm. 6.3] for the Casson-Walker invariant of rational homology spheres; see Remark 5.3 for details.…”

Link invariants via counting surfaces

“…These formulas generalize the calculation of a linking number by counting subdiagrams of special geometric-combinatorial types with signs and weights in a given link diagram. This technique is also very helpful in the rapidly developing field of virtual knot theory (see [12]), as well as in 3-manifold theory (see [17]).…”