We introduce new simplicial complexes by using various invariants and local moves for knots, which give generalizations of the Gordian complex defined by Hirasawa and Uchida. In particular, we focus on the simplicial complex defined by using the Alexander-Conway polynomial and the Delta-move, and show that the simplicial complex is Gromov hyperbolic and quasi-isometric to the real line.

Let $\mathcal{P}$ denote the set of all prime numbers, and let $p_{k}$ denothe the $k$th prime. In 1873 Mertens presented a quantitative proof of the divergence of the series $\sum_{p\in \mathcal{P}}\frac{1}{p}$ by showing the limit $B:= \lim_{x\to\infty}(\sum_{p\le x}\frac{1}{p} - \log\log x)$ exists with $B=0.26149\ldots$. In this paper we give another proof on the divergence of the above series. We prove the following

Theorem The sequence $f(n):= \sum_{k=1}^{n}\frac{1}{p_{k}} - \log\log n$, $n=2,3, \ldots$, is decreasing and bounded from below, and its limit equals the Mertens’s constant $B$.

In proofs of the first two conditions we use only classical estimations for $p_{k}$, obtained in 1939 and 1962 by Rosser and Schoenfeld.