Lifting link invariants by functors on nanophrases

Nanophrases have a filtered structure consisting of an infinite number ofcategories, 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 categorythat is actually less informative category than the link category, and so isthe quandle category. The former is called the pseudolink category and thelatter the quasilink category, which are covered by the link category; there isa known filtration: links cover pseudolinks, pseudolinks cover virtual strings,and virtual strings cover free links. The introduction and groundwork fornanophrases were done by Turaev around 2005. In this paper, we introducefunctors (Theorem 1) from general nanophrases to virtualstrings/pseudolinks/quasilinks/free links. These functors are powerful becauseof the difference between each of two. To demonstrate the effectiveness of suchnew construction of functors, the Jones pseudolink polynomial (Section 4),which implies the Jones link polynomial, is extended to general filterednanophrases using one of the functors (Section 5). In particular, theinformation level of the nanophrases used in each category is explicitlydescribed in the above construction process. This paper is a refined version bydeveloping parts of sections 3 and 6 in arXiv: 0901.3956.