In knot theory, Willerton's fish is an unexplained relationship between the first two Vassiliev invariants of a knot. These invariants are, the quadratic coefficient of the Alexander–Conway polynomial, and, an order-three invariant derived from the Jones polynomial.[1] [2]
When the values of and, for knots of a given fixed crossing number, are used as the and 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,[2] suggesting that the crossing number,, and may be related to each other by not-yet-proven inequalities.[1]
This shape is named after Simon Willerton,[1] who first observed this phenomenon and described the shape of the scatterplots as "fish-like".[3]