Lifting link invariants by functors on nanophrases

Abstract

Nanophrases have a filtered structure consisting of an infinite number of categories, and each category has a homotopy structure. Among these categories, the one that we are most familiar with is the category of links. Interestingly, the category in which the Jones polynomial is defined consists of a category that is actually less informative category than the link category, and so is the quandle category. The former is called the pseudolink category and the latter the quasilink category, which are covered by the link category; there is a known filtration: links cover pseudolinks, pseudolinks cover virtual strings, and virtual strings cover free links. The introduction and groundwork for nanophrases were done by Turaev around 2005. In this paper, we introduce functors (Theorem 1) from general nanophrases to virtual strings/pseudolinks/quasilinks/free links. These functors are powerful because of the difference between each of two. To demonstrate the effectiveness of such new construction of functors, the Jones pseudolink polynomial (Section 4), which implies the Jones link polynomial, is extended to general filtered nanophrases using one of the functors (Section 5). In particular, the information level of the nanophrases used in each category is explicitly described in the above construction process. This paper is a refined version by developing parts of sections 3 and 6 in arXiv: 0901.3956.