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Harary–Kauffman conjecture for alternating knot diagrams

Let D{\mathcal D} be an alternating knot diagram with no nugatory crossings. The determinant of D{\mathcal D} is the order of its first homology group of the double branched cover, and a Fox pp-coloring is a coloring of the arcs by elements of Z/pZ\mathbb{Z}/p\mathbb{Z} satisfying the Fox coloring rule at every crossing. Harary–Kauffman conjecture. If the determinant of D{\mathcal D} is a prime number pp, then every non-trivial Fox pp-coloring of D{\mathcal D} assigns different colors to different arcs of D{\mathcal D}. The conjecture is known for rational and Montesinos knots, and this paper proves it for the Turk's Head knots considered except when m≥5m\geq 5 is odd and n≥3n\geq 3 is relatively prime to mm; the remaining case is supported by evidence.

Sources & referencesView supporting material

Primary source

Nicholas E. Dowdall, Thomas W. Mattman, Kevin Meek and Pablo R. Solis, “On the Harary-Kauffman Conjecture and Turk's Head Knots”, arXiv:0811.0044 (2008).

alternating-knotscoloringsknot-theory

Progress summary

Refreshed
Solved

A 2009 proof settles the conjecture for all reduced alternating knot diagrams with prime determinant.

The conjecture, posed by Louis Kauffman and Frank Harary in 1999, asserts that every non-trivial Fox pp-coloring of a reduced alternating knot diagram with determinant pp uses a distinct color on every arc.

Known results

  • Pretzel and Montesinos knots: proved before the general result (Mattman and Solis, 2003).
  • Rational knots and many Turk’s Head knots: established in earlier special cases.
  • Turk’s Head knots: proved except for odd m≥5m \ge 5, n≥3n \ge 3, and gcd⁡(m,n)=1\gcd(m,n)=1 (Mattman and Solis, 2008).

July 2009 general proof

Thomas W. Mattman and Pablo Solis announced a complete proof. They show that every reduced alternating diagram with prime determinant pp has all arcs differently colored in every non-trivial Fox pp-coloring; a later source also records their stronger generalized result for prime alternating links.

Current status (as of August 2026): The original Harary–Kauffman conjecture is settled by the 2009 proof, and no sourced objection or counterexample was found.

Sources

Solutions 0

No solutions have been posted yet.