Any 7-colorable knot can be colored by four colors. (English) Zbl 1202.57010

One of the simplest invariants of a knot is its tri-colorability, which requires a knot diagram to determine. A knot diagram \(D\) for a knot \(K\subset \mathbb{R}^3\) is a collection of arcs representing the projection of \(K\) to a \(2\)-dimensional plane. Three arcs meet at any crossing, and the question of whether the arcs of \(D\) can be colored so that at each crossing three distinct colors meet, is a knot invariant. Letting the three colors correspond to elements of \(\mathbb{Z}_3\), tri-colorability is expressed as a labeling of the arcs of \(D\) such that at each crossing, the distinct labels \(a,c\), and \(b\) of the lower arcs and upper arc respectively satisfy \(a+c=2b\) in \(\mathbb{Z}_3\). Generalizing, for an odd prime \(p\), a knot diagram \(D\) is \(p\)-colorable if the arcs of \(D\) can be labeled by the elements of \(\mathbb{Z}_p\) such that distinct labels \(a,c,b\) as above at each crossing satisfy \(a+c=2b\) in \(\mathbb{Z}_p\).
L. H. Kauffman and P. Lopes [Adv. Appl. Math. 40, No. 1, 36–53 (2008; Zbl 1151.57008)] proved that if \(p>3\), a non-trivial coloring of \(D\) uses at least four colors. S. Satoh [Osaka J. Math. 46, No. 4, 939–948 (2009; Zbl 1182.57005)] proved that every \(5\)-colorable knot admits a knot diagram with exactly four colors. The main theorem of this paper proves that every \(7\)-colorable knot admits a knot diagram using exactly the four colors \(0,1,2\), and \(4\). This disproves a conjecture of Kauffman and Lopes about the minimum number of colors required for a \(p\)-coloring of certain \((2,n)\)-torus knots. The proof consists of examining the possible colorings at a crossing and using Reidemeister moves to introduce new crossings and adjusting the labels to eliminate various colors. The author also proves the same result for \(7\)-colorable ribbon \(2\)-knots (a ribbon \(2\)-knot is obtained from trivially linked \(2\)-spheres embedded in \(\mathbb{R}^4\) by attaching a \(1\)-handle; the corresponding knot diagram is a projection to a \(3\)-dimensional plane).


57M25 Knots and links in the \(3\)-sphere (MSC2010)
05C10 Planar graphs; geometric and topological aspects of graph theory
05C15 Coloring of graphs and hypergraphs
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