On the First Two Vassiliev Invariants

The two simplest nontrivial Vassiliev knot invariants (see [Vassiliev 92, Birman and Lin 93]) are of type two and type three. These invariants have been studied from various angles: for instance, combinatorial formulæ for evaluating them have been derived, and simple bounds in terms of crossing number have been obtained (see e.g., [Polyak and Viro 94, Lannes 93, Willerton 97]). In this work, the invariants are examined from the novel perspective of the actual values that they take on knots of small crossing number. For instance, one can ask how accurate the known bounds are, as in Section 2. When looking at this question I plotted the values of these invariants which revealed the interesting “fish” plots in Section 3: these pictures form the focus of this paper. Various questions arising from these graphs can be answered for torus knots (see Section 4). Section 5 presents some problems and further questions.