Invariant Trace Fields of Chain Links

In this paper, we compute the trace field of C(2, s), the complement of two component chain link with s left half twists in ${\mathbb{S}}^3$, for every s. As a result, for every $n{\in}{\mathbb{N}}{\backslash}\{1\}$, we can find $s{\in}{\mathbb{Z}}$ such that the degree of the trace field of C(2, s) is n. We also prove that if for fixed p, the degree of the trace field of C(p, s) runs over ${\mathbb{N}}{\backslash}\{1\}$, then p is contained in {1, 2, 4, 8}.