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Detecting the orientation of string links by finite type invariants. (English) Zbl 1159.57003

Funct. Anal. Appl. 41, No. 3, 208-216 (2007); translation from Funkts. Anal. Prilozh. 41, No. 3, 48-59 (2007).
In general, knot polynomials as well as quantum invariants cannot distinguish a knot from its inverse. Although the case of Vassiliev invariants is still open, it is conjectured to be negative. There are a few papers for the case of links. In particular, Xiao-Song Lin [Enseign. Math., II. Sér. 47, No. 3–4, 315–327 (2001; Zbl 1002.57033)] showed that Vassiliev invariants distinguish links with at least six components from their inverses. Here, the inverse of a link is the link obtained by reversing the orientations of all components.
The paper under review deals with the problem of detecting the orientation by finite type invariants for string (equivalently, long) links. It is easily shown that there exists a Vassiliev invariant which detects the orientation for string links with more than two components. Thus only the case of string links with two components is non-trivial. The main result claims that there exists a Vassiliev invariant of degree \(7\) and a two-component string link such that the invariant takes distinct values on the link and its inverse. Bar-Natan’s table on his website shows that \(7\) is the smallest degree of a Vassiliev invariant that can detect the orientation of two-component string links.
Two proofs of this result are given in the paper. The invertibility problem of \(p\)-component string links is related to the problem of whether the algebra of chord diagrams on \(p\) strings is commutative. In fact, the non-commutativity of the algebra is obvious for \(p>2\) and folklore for \(p=2\), but the latter case never appeared in literature. Two chord diagrams are exhibited for \(p=2\), and the proof involves a computer calculation. Its data is available at the website of the first author. The second proof uses Jacobi diagrams and the calculation can be done by hand. There is a remark that the smallest diagram in degree \(7\) was first discovered by Bar-Natan (not published).

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)

Citations:

Zbl 1002.57033
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References:

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