knot theory In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ...

, Willerton's fish is an unexplained relationship between the first two

Vassiliev invariant In the knot theory, mathematical theory of knots, a finite type invariant, or Vassiliev invariant (so named after Victor Anatolyevich Vassiliev), is a knot invariant that can be extended (in a precise manner to be described) to an invariant of cert ...

s of a

knot A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...

. These invariants are , the quadratic

coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...

of the

Alexander–Conway polynomial In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a ...

, and , an order-three invariant derived from the

Jones polynomial In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polyno ...

... See in particular Section 4.2.1, "Willerton's fish and families of knots". When the values of and , for knots of a given fixed crossing number, are used as the and coordinates of a

scatter plot A scatter plot, also called a scatterplot, scatter graph, scatter chart, scattergram, or scatter diagram, is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of dat ...

, 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 curve In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an eq ...

s, suggesting that the crossing number, , and may be related to each other by not-yet-proven

inequalities Inequality may refer to: * Inequality (mathematics), a relation between two quantities when they are different. * Economic inequality, difference in economic well-being between population groups ** Income inequality, an unequal distribution of i ...

. This shape is named after Simon Willerton, who first observed this phenomenon and described the shape of the scatterplots as "fish-like"..

References

{{reflist Knot theory