Willerton's fish

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.[1][2]

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,[2] suggesting that the crossing number, c2, and j3 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]

References
  1. Chmutov, S.; Duzhin, S.; Mostovoy, J. (2012), "14.3 Willerton's fish and bounds for c2 and j3", Introduction to Vassiliev knot invariants (PDF), Cambridge University Press, Cambridge, pp. 419–420, arXiv:1103.5628, doi:10.1017/CBO9781139107846, ISBN 978-1-107-02083-2, MR 2962302.
  2. Dunin-Barkowski, P.; Sleptsov, A.; Smirnov, A. (2013), "Kontsevich integral for knots and Vassiliev invariants", International Journal of Modern Physics A, 28 (17): 1330025, arXiv:1112.5406, Bibcode:2013IJMPA..2830025D, doi:10.1142/S0217751X13300251, MR 3081407. See in particular Section 4.2.1, "Willerton's fish and families of knots".
  3. Willerton, Simon (2002), "On the first two Vassiliev invariants", Experimental Mathematics, 11 (2): 289–296, MR 1959269.
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