## 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).

### MSC:

 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

### Keywords:

knots; knot colorings; 7-colorings; colors; ribbon 2-knots

### Citations:

Zbl 1151.57008; Zbl 1182.57005
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### References:

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