What is the easiest construction of the Casson invariant? The original construction using representation spaces (as found, for instance, in Akbulut-McCarthy) is very technical since you have to perturb things to make them transverse.


One possibility would be to do some kind of Kirby calculus construction. Such a thing can be found in Walker's book and Lescop's book, but they constructed much more general things so their work is also very technical.



Construction of the Casson invariant
Hoste has a combinatorial formula for the Casson invariant for integral homology $3$-spheres. Let $\Sigma$ be an integral homology $3$-sphere obtained via surgery on a framed link $L = K\_1 \cup \dots \cup K\_n$ in $S^3$ with framings $1/q\_i$. Additionally, we assume $\mathrm{lk}(K\_i, K\_j) = 0$ for $i \neq j$ (this can always be arranged, see Lemma 12.2 in Saveliev's *Lectures on the Topology of $3$-Manifolds*. In fact, that lemma says that we can take $q\_i = \pm 1$). We can write the Conway polynomial of $L$ in the form
$$\nabla\_L(z) = z^{n-1}(a\_0(L) + a\_1(L) z^2 + \cdots + a\_m(L) z^{2m}).$$
With the above notations, Hoste's formula is
$$\lambda(\Sigma) = \sum\_{L' \subset L} \left( \prod\_{i : K\_i \subset L'} q\_i\right) a\_1(L').$$
The sum is taken over all sublinks $L'$ of $L$.


You can see the details of Hoste's formula in his original paper:



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> Hoste, Jim. [*A formula for Casson's invariant*](http://www.ams.org/journals/tran/1986-297-02/S0002-9947-1986-0854084-4/). Trans. Amer. Math. Soc. 297 (1986), 547-562
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Hoste's original paper does not show that $\lambda$ as defined by his formula is actually an invariant of integral homology $3$-spheres, however. The missing ingredient is a Kirby move that preserves the unlinking of all components of $L$ as required for Hoste's formula to be stated. Such a move is constructed by Habiro in the following paper:



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> Habiro, Kazuo. [*Refined Kirby calculus for integral homology spheres*](http://www.msp.warwick.ac.uk/gt/2006/10/p031.xhtml). Geometry & Topology 10 (2006), 1285–1317.
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Combining the results of the above two papers gives a nice combinatorial construction of the Casson invariant as an invariant of integral homology $3$-spheres.


