knot theory In the mathematical field of 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 ...

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

Vassiliev invariant Vasilyev, Vasiliev or Vassiliev or Vassiljev (russian: Васильев), or Vasilyeva or Vasilieva (feminine; russian: link=no, Васильева), is a common Russian surname that is derived from the Russian given name ''Vasiliy'' (equivalent o ...

s of a

knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...

. These invariants are , the quadratic

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...

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 ver ...

, 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 polynom ...

... 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 data. ...

, 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 equ ...

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

inequalities Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...

. 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