The fourth skein module of three-dimensional manifolds

Skein modules are the main objects of an algebraic topology based on knots. They are quotients of free modules over ambient isotopy classes of links in a 3-manifold by properly chosen local (skein) relations. There are as many skein modules as the number of skein relations. Here we concentrate on the concept of the nth skein modules ( n ≥ 2). The nth skein module is a skein module based on the skein relation b₀ L₀ + b₁L ₁ + ··· + b_{n −1}L_n−1 = 0 and a framing relation L(1) = aL (The framing relation is necessary in the case n ≥ 3). The second skein module is relatively easy to analyze and the third skein module is interesting to study. In fact, the third skein module gives the Kauffman bracket skein module in the case of unoriented links and gives the HOMFLYPT skein module in the case of oriented links.; We analyze the concept of the fourth skein module of 3-manifolds, that is a skein module of unoriented links based on the skein relation b₀L₀ + b ₁L₁ + b₂ L₂ + b₃L ₃ = 0 (b₀, b₃ invertible) and a framing relation L(1) = aL. We give necessary conditions for trivial links to be linearly independent in the module. In the case of classical links (i.e. links in S3) 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). 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 an n-algebraic tangle (and link) and analyze the skein module for 3-algebraic and 4-algebraic links. As a byproduct we prove the Montesinos-Nakanishi 3-moves conjecture for 3-algebraic links (including 3-bridge links) and 4-bridge links. Moreover we study a relationship between 4-algebraic links and the Montesinos-Nakanishi conjecture. In fact, we show the double of Borromean ring (6 braid link) and Chen's link (5 braid link) are both 4-algebraic links.