We twist a knot by twisting some solid tori, each of which contains the knot, one by one to the meridional direction. Suppose the solid tori have mutually disjoint essential toral boundaries in the knot exterior.

M. Kouno, K. Motegi and T. Shibuya have shown that, by twisting a knot along one or two solid tori in an inclusion relation, the resulting knot type is different from the original one (see [5] and [6]). In this paper, we investigate whether twisting along various solid tori also produces a different knot type or not.

The linear homogeneous differential equation $z''=12yz$ with $y$ an arbitrary solution of Painlevé's first equation $y''=6y^2+x$ will be proved irreducible.

For an outer action $\alpha$ of the symmetric group $S_3$ on an AFD factor $M$, we will determine when the pair $M\rtimes_\alpha S_3 \supset M\rtimes_\alpha S_2$ contains a Cartan subalgebra.

We find a formula for the $Q$-polynomial of a pretzel link $L(p_1,p_2,\ldots,p_n)$. The formula describes the $Q$-polynomial as a polynomial in $H_{p_1}(x),H_{p_2}(x),\ldots,H_{p_n}(x)$, where $H_{p_1}(x),H_{p_2}(x),\ldots,H_{p_n}(x)$ are Laurent polynomials in $x$ and $H_{p_i}(x)$ depends on only $p_i$.

Let $f$ be a density function with respect to Lebesgue measure. We suppose that $f(x)>0$ on $(0,\beta)$, where $0<\beta\leqq+\infty$, and $f$ is uniformly continuous on $(0,\beta)$. Moreover, let $f'(x)\to\alpha$ as $x\to +0$ exist, where $0<\alpha<+\infty$. We consider a non-regular model defined by $f(x,\theta)=f(x-\theta)$, $\theta,x\in\mathbf{R}$. In the present paper, under some conditions, it is shown that when $\theta$ is regarded as a random variable with a prior density function with respect to Lebesgue measure, there exist asymptotic expansions of centered and scaled posterior distributions of $\theta$.