Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.4

Sequences of Spiral Knot Determinants

Seong Ju Kim, Ryan Stees, and Laura Taalman
Department of Mathematics and Statistics
James Madison University
305 Roop Hall, MSC 1911
Harrisonburg, VA 22807


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.

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(Concerned with sequences A000027 A000578 A001353 A001834 A001906 A002878 A004146 A005013 A006235 A007877 A098149 A108412 A121022 A131022 A131027 A251610.)

Received March 10 2015; revised versions received October 18 2015; November 17 2015. Published in Journal of Integer Sequences, December 17 2015.

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