Document Zbl 0961.57006 - zbMATH Open
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On Gusarov’s groups of knots. (English) Zbl 0961.57006
Summary: We give a construction of Gusarov’s groups \({\mathcal G}_n\) of knots based on pure braid commutators, and show that any element of \({\mathcal G}_n\) is represented by an infinite number of prime alternating knots of braid index less than or equal to \(n+1\). We also study \({\mathcal V}_n\), the torsion-free part of \({\mathcal G}_n\), which is the group of equivalence classes of knots which cannot be distinguished by any rational Vassiliev invariant of order less than or equal to \(n\). Generalizing the Gusarov-Ohyama definition of \(n\)-triviality, we give a characterization of the elements of the \(n\)-th group of the lower central series of an arbitrary group.
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |