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 $c 2$ , the quadratic coefficient of the Alexander–Conway polynomial, and $j 3$ , an order-three invariant derived from the Jones polynomial.^[1]^[2]

When the values of $c 2$ and $j 3$ , 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, $c 2$ , and $j 3$ 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

^ ^a ^b ^c Chmutov, S.; Duzhin, S.; Mostovoy, J. (2012), "14.3 Willerton's fish and bounds for $c 2$ and $j 3$ ", 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.
^ ^a ^b 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".
^ Willerton, Simon (2002), "On the first two Vassiliev invariants", Experimental Mathematics, 11 (2): 289–296, MR 1959269.

[cdm-1] Chmutov, S.; Duzhin, S.; Mostovoy, J. (2012), "14.3 Willerton's fish and bounds for $c 2$ and $j 3$ ", 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.

[dss-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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Willerton's fish

References

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