Spiral knots are a generalization of the well-known class of torus knots indexed by strand number and base word repetition. By fixing the strand number and varying the repetition index, we obtain integer sequences of spiral knot determinants. In this paper, we examine such sequences for spiral knots of up to four strands using a new periodic crossing matrix method. Surprisingly, the resulting sequences vary widely in character and, even more surprisingly, nearly every one of them is a known integer sequence in the Online Encyclopedia of Integer Sequences. We also develop a general form for these sequences in terms of recurrence relations that exhibits a pattern which is potentially generalizable to all spiral knots.