Montesinos–Nakanishi conjecture and its basic-tangle generalizations

Let $L$ be a link, possibly a relative link, in a $3$-manifold $M$. An $n$-move is a local change that adds $n$ positive or negative half-twists, and two links are $3$-equivalent if they are related by $3$-moves and their inverses.

Montesinos–Nakanishi conjecture. Every link is $3$-equivalent to a trivial link. More generally, every $2$-tangle is $3$-equivalent to a tangle with at most one crossing; every $3$-tangle is $3$-equivalent to one of the $40$ basic $3$-tangles, with possible trivial components; and there is a finite number $g(n)$ of basic $n$-tangles such that every $n$-tangle is $3$-equivalent to one of them, with possible trivial components. The conjectured number is

$$g(n)=\prod_{i=1}^{n-1}(3^i+1),$$

so $g(4)=1120$.

The first assertion is the Montesinos–Nakanishi conjecture; the paper states that the assertions for $2$- and $3$-algebraic tangles are proved later. The conjecture concerns finite generation of tangles and links under $3$-moves.