Casson invariant

Builds on Homology sphere Knot Dehn surgery Alexander polynomial

Definition

Let $\Sigma$ be an oriented integral homology 3-sphere, let $K \subset \Sigma$ be a knot and let $n$ be an integer. Let $\Sigma_{K, 1/n}$ be the result of Dehn surgery on $K$ with coefficient $1/n$ ; it is again an oriented integral homology $3$ -sphere. Let $\Delta_K(t)$ be the Alexander polynomial of $K$ , normalised so that $\Delta_K(t^{-1}) = \Delta_K(t)$ and $\Delta_K(1) = 1$ .

The Casson invariant is the unique function $\lambda$ from oriented integral homology $3$ -spheres, up to orientation-preserving homeomorphism, to $\Z$ such that:

$\lambda(S^3) = 0$ , and
for every $\Sigma$ , $K$ and $n$ as above, $\lambda(\Sigma_{K, 1/(n+1)}) - \lambda(\Sigma_{K, 1/n}) = \tfrac{1}{2} \Delta_K''(1).$

Casson proved that such a function exists and is unique.

Remarks

Example: for a trefoil knot $K$ in $S^3$ , $\Delta_K(t) = t - 1 + t^{-1}$ , so $\Delta_K''(t) = 2 t^{-3}$ and $\tfrac12 \Delta_K''(1) = 1$ . Since $S^3_{K, 1/0} = S^3$ , clause 2 with $n = 0$ gives $\lambda(S^3_{K, 1}) = 1$ . For the right-handed trefoil, $S^3_{K, 1}$ is the Poincaré homology sphere with the orientation opposite to that of $\Sigma(2, 3, 5)$ , and $\lambda(\Sigma(2, 3, 5)) = -1$ .

Further properties: $\lambda(-\Sigma) = -\lambda(\Sigma)$ , $\lambda$ is additive under connected sum, and $\lambda(\Sigma)$ is congruent modulo $2$ to the Rokhlin invariant of $\Sigma$ .

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Alexander polynomial concept
Dehn surgery concept
Homology sphere concept
Knot concept
Casson invariant definition

Tag	`FWF9D69R3F`
URI	`m2://FWF9D69R3F`
Kind	definition
Writing	W0 stub
MSC	57K31
Defines	Casson invariant
Uses	Homology sphere, Knot, Dehn surgery, Alexander polynomial
Changed	2 November 2026

Casson invariant

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here