For a fixed isotopy type K of unframed knots in S there are infinitely many isotopy classes of framed knots that correspond to K when we forget the framing. We show that the same fact is true for all the isotopy types of unframed knots in a closed oriented 3-manifold M , provided that M 6= (S × S)#M . On the other hand for any M = (S × S)#M ′ we construct examples of isotopy classes of unframed...

We continue the study of the equivalence relations introduced in the first part. For finite k, it is shown that k-quasi-isotopy implies (k + 1)-cobordism of Cochran–Orr, leading to invariance of Cochran’s derived invariants β, i ≤ k. Furthermore, if two links are k-quasi-isotopic then they cannot be distinguished by any Vassiliev invariant of type ≤ k which is well-defined up to PL isotopy, whe...

The concept of Smarandache isotopy is introduced and its study is explored for Smarandache: groupoids, quasigroups and loops just like the study of isotopy theory was carried out for groupoids, quasigroups and loops. The exploration includes: Smarandache; isotopy and isomorphy classes, Smarandache f, g principal isotopes and G-Smarandache loops.

The concept of Smarandache isotopy is introduced and its study is explored for Smarandache: groupoids, quasigroups and loops just like the study of isotopy theory was carried out for groupoids, quasigroups and loops. The exploration includes: Smarandache; isotopy and isomorphy classes, Smarandache f, g principal isotopes and G-Smarandache loops.

In this paper, we introduce a method to construct ambient isotopy invariants for smooth imbeddings of closed surfaces into 4-space by using hyperbolic splittings of the imbedded surfaces and an arbitrary given isotopy or regular isotopy invariant of classical knots and links in 3-space. Using this construction, adopting the Kauffman bracket polynomial as an example, we produce some invariants.

SERGEY A. MELIKHOV and DUŠAN REPOVŠ ABSTRACT We prove that k-quasi-isotopy implies (k + 1)-cobordism of Cochran–Orr, leading to k-quasi-isotopy invariance of Cochran's derived invariants β i , i ≤ k, and Milnor's ¯ µ-invariants of length ≤ 2k + 3. Secondly, k-quasi-isotopic links cannot be distinguished by any Vassiliev invariant of type ≤ k which is well-defined up to PL isotopy, where type ≤ ...

Centreless Lie tori have been used by E. Neher to construct all extended affine Lie algebras (EALAs). In this article, we study isotopy for centreless Lie tori, and show that Neher’s construction provides a 1-1 correspondence between centreless Lie tori up to isotopy and families of EALAs up to isomorphism. Also, centreless Lie tori can be coordinatized by unital algebras that are in general no...

The concept of Smarandache isotopy is introduced and its study is explored for Smarandache: groupoids, quasigroups and loops just like the study of isotopy theory was carried out for groupoids, quasigroups and loops. The exploration includes: Smarandache; isotopy and isomorphy classes, Smarandache f, g principal isotopes and G-Smarandache loops.