## Properties of minimal charts and their applications. II.(English)Zbl 1194.57030

Charts were introduced by S. Kamada [J. Knot Theory Ramifications 1, No. 2, 137–160 (1992; Zbl 0763.57013)] as a tool for representing surface braids. A chart is an oriented planar graph $$\Gamma$$ with edges labeled by natural numbers, satisfying certain conditions. In particular, the vertices of $$\Gamma$$ are allowed to have degree 1 (black vertices), degree 4 (crossings), and degree 6 (white vertices). Moreover, the edges converging at any white vertex are alternately labeled by $$i$$ and $$i+1$$ for some $$i \geq 1$$, and we call that vertex an $$i$$-vertex. Two $$n$$-charts represent the same surface braid if and only if they are equivalent up to certain local modifications called $$C$$-moves.
The complexity of a chart $$\Gamma$$ is the pair $$(w(\Gamma),-f(\Gamma))$$, where $$w(\Gamma)$$ is the number of white vertices and $$f(\Gamma)$$ is the number of edges joining two black vertices. Then $$\Gamma$$ is called a minimal chart if it has minimal complexity among the charts $$C$$-move equivalent to it with respect to the lexicographic order. It is known that there is no minimal chart $$\Gamma$$ with $$w(\Gamma) = 1,2,3,5$$, while there exists such a minimal chart with $$w(\Gamma) = 4,6$$.
In this paper, the author continue their investigation on minimal charts, started with [J. Math. Sci., Tokyo 14, No. 1, 69–97 (2007; Zbl 1135.57012)] and aimed to prove that there are no minimal charts with $$w(\Gamma) = 7$$.
The main result here is that if $$\Gamma$$ is a minimal chart with $$w(\Gamma) = 7$$ then the sequence $$(n_1, \dots, n_k)$$, with $$n_i$$ the number of white $$i$$-vertices of $$\Gamma$$, is one of the following, up to translation or reversion of the labeling: $$(7)$$, $$(5,2)$$, $$(4,3)$$, $$(3,2,2)$$, $$(2,3,2)$$.

### MSC:

 57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)

### Keywords:

2-knot; surface braid; minimal chart

### Citations:

Zbl 0763.57013; Zbl 1135.57012