Rational invariants of finite type fail to distinguish knots in \(\mathbb{S}^2\times \mathbb{S}^1\) (Q2708171)

The author constructs some knots in \(S^2\times S^1\) that cannot be distinguished by rational invariants of finite type in the sense of Vassiliev. But some of them can be distinguished by invariants of finite type with values in a finite abelian group. The key point is the construction of geometric sequences of knots: if a geometric sequence of braids is a sequence \(\tau^z\) , where \(z\) is in \(\mathbb{Z}\) and \(\tau\) is a pure braid, then a geometric sequence of knots \(K_z\), where \(z\) is in \(\mathbb{Z}\), in a manifold \(M\) is a sequence of knots in \(M\) which are the same except within a ball \(B\subset M\) where they differ as a geometric sequence of braids. The author constructs some periodic sequences of knots and proves that some knots in these sequences are different knots that cannot be distinguished by rational invariants of finite type in the sense of Vassiliev.