In the 1990s, Habiro defined Ck-move of oriented links for each natural number k
[5]. A Ck-move is a kind of local move of oriented links, and two oriented knots have
the same Vassiliev invariants of order [les ] k−1 if and only if they are transformed into
each other by Ck-moves. Thus he has succeeded in deducing a geometric conclusion
from an algebraic condition. However, this theorem appears only in his recent paper
[6], in which he develops his original clasper theory and obtains the theorem as a
consequence of clasper theory. We note that the ‘if’ part of the theorem is also shown
in [4], [9], [10] and [16], and in [13] Stanford gives another characterization of knots
with the same Vassiliev invariants of order [les ] k−1.
Master integrals for bipartite cuts of three-loop photon self energy
