The Fourth Skein Module and the Montesinos-nakanishi Conjecture for 3-algebraic Links

We study the concept of the fourth skein module of 3-manifolds, that is a skein module based on the skein relation b0L0 + b1L1 + b2L2 + b3L3 = 0 and a framing relation L = aL (a, b0, b3 invertible). We give necessary conditions for trivial links to be linearly independent in the module. We investigate the behavior of elements of the skein module under the n-move and compute the values for (2, n)-torus links and twist knots as elements of the skein module. Using the idea of mutants and rotors, we show that there are different links representing the same element in the skein module. We also show that algebraic links (in the sense of Conway) and closed 3-braids are linear combinations of trivial links. We introduce the concept of n-algebraic tangles (and links) and analyze the skein module for 3algebraic links. As a byproduct we prove the Montesinos-Nakanishi 3-moves conjecture for 3-algebraic links (including 3-bridge links). In the case of classical links (i.e. links in S) our skein module suggests three polynomial invariants of unoriented framed (or unframed) links. One of them generalizes the Kauffman polynomial of links and another one can be used to analyze amphicheirality of links (and may work better than the Kauffman polynomial). In the end, we speculate about the meaning and importance of our new knot invariants.