Invariants of colored links.

*(English)*Zbl 0758.57004The authors define a family of multivariable invariants of oriented links, one for each integer \(N\geq 2\), with one variable associated to each component of the link. The integer \(N\) is the dimension of a vector space \(V\) used in the construction. This construction depends on representing the category of oriented tangles on the category of vector spaces and linear maps, by means of a suitable choice of representative for the elementary tangles. The variables are incorporated appropriately as scalars; an \(N\)th root of unity is also used in the definitions, and the underlying vector spaces are treated as complex spaces.

An essential part of the definition of the link invariant is the use of a \((1,1)\)-tangle rather than a \((0,0)\)-tangle to represent a given link, as the construction given always yields the zero map as the direct representation of any \((0,0)\)-tangle. This property ensures that for each \(N\) the invariant behaves like the Alexander polynomial in that it vanishes on split links.

The authors note that J. Murakami has established the equivalence of the multivariable Alexander polynomial with this invariant in the case \(N=2\). It is possible that the other cases are also related to the multivariable Alexander polynomial; no explicit examples appear to be available which might show that the invariants for \(N>2\) are more powerful than the Alexander polynomial.

An essential part of the definition of the link invariant is the use of a \((1,1)\)-tangle rather than a \((0,0)\)-tangle to represent a given link, as the construction given always yields the zero map as the direct representation of any \((0,0)\)-tangle. This property ensures that for each \(N\) the invariant behaves like the Alexander polynomial in that it vanishes on split links.

The authors note that J. Murakami has established the equivalence of the multivariable Alexander polynomial with this invariant in the case \(N=2\). It is possible that the other cases are also related to the multivariable Alexander polynomial; no explicit examples appear to be available which might show that the invariants for \(N>2\) are more powerful than the Alexander polynomial.

Reviewer: H.R.Morton (Liverpool)

##### MSC:

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