In knot theory, Willerton's fish is an unexplained relationship between the first two Vassiliev invariants of a knot. These invariants are c2, the quadratic coefficient of the Alexander–Conway polynomial, and j3, an order-three invariant derived from the Jones polynomial.
When the values of c2 and j3, for knots of a given fixed crossing number, are used as the x and y coordinates of a scatter plot, the points of the plot appear to fill a fish-shaped region of the plane, with a lobed body and two sharp tail fins. The region appears to be bounded by cubic curves, suggesting that the crossing number, c2, and j3 may be related to each other by not-yet-proven inequalities.
This shape is named after Simon Willerton, who first observed this phenomenon and described the shape of the scatterplots as "fish-like".