On the minimum number of colors for knots

Louis H Kauffman,Pedro Lopes

doi:10.1016/j.aam.2006.11.006

Louis H Kauffman, Pedro Lopes

Open Access

https://doi.org/10.1016/j.aam.2006.11.006

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Abstract

In this article we take up the calculation of the minimum number of colors needed to produce a non-trivial coloring of a knot. This is a knot invariant and we use the torus knots of type ( 2 , n ) as our case study. We calculate the minima in some cases. In other cases we estimate upper bounds for these minima leaning on the features of modular arithmetic. We introduce a sequence of transformations on colored diagrams called Teneva transformations. Each of these transformations reduces the number of colors in the diagrams by one (up to a point). This allows us to further decrease the upper bounds on these minima. We conjecture on the value of these minima. We apply these transformations to rational knots.

Highlights

The colorings we are concerned with are the so-called Fox colorings, [4, 7]
We prove that two distinct colors are not enough to produce a non-trivial coloring in the n, r = 3 case
In order to prove that k + 2 is an upper bound, as referred to above, we consider the diagram of the torus knot T (2, 2k + 1) as given by the braid closure of σ12k+1 (σ1 ∈ B2) and endowed with a non-trivial (2k + 1)-coloring

Summary

Introduction

The colorings we are concerned with are the so-called Fox colorings, [4, 7]. Given a knot diagram and an integer r we consider the integers 0, 1, 2, . . . , r − 1 mod r, whose set will be denoted Zr. We will obtain diagrams endowed with non-trivial colorings that use less colors than the ones considered so far, these diagrams have more arcs than the σ1n’s.

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