I have this question but I have no idea how to begin. Could someone give a solution for K_1?
2026-03-26 14:40:46.1774536046
Vassiliev invariant calculation
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(I am hesitant answering this question, especially since the asker seems to have disappeared, but I need some practice calculating with finite-type invariants anyway.)
The definition of $v_2$ is specifying its value using chord diagram. The chords specify which points are the double points for the map $S^1\to S^3$. For $K_1$, the following singular knots are useful:
Expanding the first diagram gives
where the second term in the first expansion is $0$ since its two expansions are unknots and cancel out. Thus, $v_2$ of the right-handed trefoil is $a+b$.