Equivalence of Links Let $L,J$ be two equivalent links. Let $L_1,...,L_n$ be the components of $L$ and $J_1,
...,J_m$ the components of $J$. Is it necessarily true that $n = m$ and each $L_i$ is equivalent to a distinct $J_k$ as knots?
I believe $n = m$ because I cannot see how Reidemeister moves can change the number of components, but I'm not completely convinced with the second statement. Any insight is appreciated.
 A: Label each of the $n$ components of $L$ with distinct numbers $1,\dots,n$.  Apply Reidemeister moves that show $L$ is equivalent to $J$, keeping track of the labels -- analyzing each of the three Reidemeister moves separately, it's clear that the labels can be preserved through each move.  In particular, components with different labels never merge and components never split into two components.
At the end, the labels can be used to label the components of $J$.  By the analysis, each component gets a distinct label, and since this is an equivalence of links all components get a label.  Since the $n$ labels can label both $L$ and $J$, they each have $n$ components.  Furthermore, the labels form a bijection from the components of $L$ to the components of $J$ with the property that each component is mapped to an equivalent component, since we can erase individual components from entire sequences of Reidemeister moves to get new sequences of Reidemeister moves.
Warning: The converse isn't true: just because two links have a bijection between their components such that each component maps to an equivalent knot, it doesn't mean the links are equivalent.  There are infinitely many inequivalent two-component links whose components are unknots.
