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In this paper we compute the trace field for the family of hyperbolic twist knots. We describe this field as a simple extension ℚ(z0) where z0 is a specified root of a particular irreducible polynemial Φn(z)∈ℤ[z]. As a consequence, we find that the degree of the trace field is precisely two less than the-minimal crossing number of a twist knot.
Hoste, Jim, and Patrick D. Shanahan. “Trace Fields of Twist Knots.” Journal of Knot Theory & Its Ramifications, vol. 10, no. 4, June 2001, p. 625-639. doi:10.1142/S0218216501001049.