Invisible Knots and Rainbow Rings: Knots Not Determined by Their Determinants. Issue 1 (1st January 2020)

• Record Type: Journal Article
• Authors: Ho, Nancy; Godzik, James; Jones, Jennifer; Mattman, Thomas W.; Sours, Dan
• Is Part Of: Mathematics magazine. Volume 93:Issue 1(2020)
• Page Start: 4
• Page End: 18
• Publication Date: 2020-01-01
• Subjects: MSC: 57M25; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/0025570X.2020.1685320
• Languages: English
• ISSNs: 0025-570X

Abstract

Summary: We determine p -colorability of the paradromic rings. These rings arise by generalizing the experiment of bisecting a Möbius strip. Instead of joining the ends with a single half twist, use m half twists, and, rather than bisecting ( n = 2), cut the strip into n sections. Replacing each thin strip with its midline results in the m, n paradromic link. Using the notion of p -colorability from knot theory, we determine, for each m and n, which primes p can be used to color the link. Amazingly, almost all admit 0, 1, or an infinite number of prime colorings! This is reminiscent of solutions sets in linear algebra. Indeed, the problem quickly turns into a study of the eigenvalues of a large, nearly diagonal matrix.