Band description of knots and Vassiliev invariants

Abstract

In 1993 K. Habiro defined C k -move of oriented links and around 1994 he proved that two oriented knots are transformed into each other by C k -moves if and only if they have the same Vassiliev invariants of order ≤ k−1. In this paper we define Vassiliev invariant of type (k 1 , ..., k l ), and show that, for k = k 1 + · · · + k l , two oriented knots are transformed into each other by C k -moves if and only if they have the same Vassiliev invariants of type (k 1 , ..., k l ). We introduce a concept 'band description of knots' and give a diagram-oriented proof of this theorem. When k 1 = · · · = k l = 1, the Vassiliev invariant of type (k 1 , ..., k l ) coincides with the Vassiliev invariant of order ≤ l − 1 in the usual sense. As a special case, we have Habiro's theorem stated above.

Self delta-equivalence for links whose Milnor’s isotopy invariants vanish

“…By arguments similar to those in the proof of Theorem 1.2 in [20], we have the following lemma. In [20], the authors proved it with using 'band description' defined by K. Taniyama and the author [29]. Here we give a proof with using clasper.…”

An unknotting theorem for delta and sharp edge‐homotopy

“…2.1. This move was introduced by Habiro as a local move on an oriented link [4], [2], and it was extended to spatial graphs by Taniyama and Yasuhara from a stand point of the "band description" [26] (see also [25], [19]). We note that the original definition of the C k -move is different from the one above, but it is known that each of the original C k -moves can be realized by local moves as illustrated in Fig.…”

“…A (k + 1)-component Milnor link is one of the C k−1 -links, and it is known the following. We refer the reader to [25], [19], [26] for details. Lemma 2.2.…”

Knots With Given Finite Type Invariants and Conway Polynomial

“…A double of a C k -move is called a C k+1 -move. For details, refer to [18] or [23]. To describe the position of double points, the notion of a chord diagram is introduced in [3].…”

“…M. N. Goussarov([4]) and K. Habiro ([5], [6]) showed independently that two knots can be transformed into each other by a finite sequence of standard C n -moves if and only if they have the same Vassiliev invariants of order less than n. C n -moves are originally defined by Habiro in [5]. In [18] and [23], they are defined as a family of local moves. It is known that any kind of C n -moves can be realized by a finite sequence of standard C n -moves.…”