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Abstract: Kauffman-s bracket is an invariant of regular isotopy of knots and linkswhich since its discovery in 1985 it has been used in many differentdirections: a it implies an easy proof of the invariance of in fact, it isequivalent to the Jones polynomial; b it is the basic ingredient in acompletely combinatorial construction for quantum 3-manifold invariants; c byits fundamental character it plays an important role in some theories inPhysics; it has been used in the context of virtual links; it has connectionswith many objects other objects in Mathematics and Physics. I show in this notethat, surprisingly enough, the same idea that produces the bracket can beslightly modified to produce algebraically stronger regular isotopy and ambientisotopy invariants living in the quotient ring $R-I$, where the ring $R$ andthe ideal $I$ are: \begin{center} $R=\Z\alpha,\beta,\delta$, $I=< p 1, p 2>$, with $p 1=\alpha^2 \delta + 2 \alpha \beta \delta ^2 -\delta ^2+\beta ^2\delta, p 2=\alpha \beta \delta^3+\alpha ^2 \delta ^2+\beta^2 \delta ^2+\alpha \beta\delta -\delta.$ \end{center} It is easy to prove that any pair of linksdistinguished by the usual bracket is also distinguishable by the newinvariant. The contrary is not necessarily true. However, a explicit example ofa pair of knots not distinguished by the bracket and distinguished by this newinvariant is an open problem.

Author: Sostenes Lins

Source: https://arxiv.org/

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