In this paper we describe a difference sequence technique, hereafter referred to as the twist sequence technique, for studying Vassiliev invariants. This technique is used to show that Vassiliev invariants have polynomial growth on certain sequences of knots. Restrictions of Vassiliev invariants to the sequence of (2, 2i + 1) torus knots are characterized. As a corollary it is shown that genus, crossing number, signature, and unknotting number are not Vassiliev invariants. This characterization also determines the topological information about (2, 2i + 1) torus knots encoded in finite-type invariants. The main result obtained is that the complement of the space of Vassiliev invariants is dense in the space of all numeric knot invariants. Finally, we show that the uniform limit of a sequence of Vassiliev invariants must be a Vassiliev invariant.
An algorithm that produces polygonal cable links is described, and applications discussed. In particular, the stick numbers of Tp,q torus links are shown to be 4p for 2p < q ≤ 3p, and it is shown that, in general, [Formula: see text]. Further, it is shown that the Ramsey number of a link is at least the sum of its arc index and bridge number. Using these results, we relate the Ramsey, stick and crossing numbers of torus links, showing [Formula: see text].
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