Divide knots and links, defined by A'Campo in the singularity theory of complex curves,
is a method to present knots or links by real plane curves. The present paper is a continuation
of the author's previous result that every knot in the major subfamilies of Berge's lens space surgery
(i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped curve as a divide knot.
In the present paper, L-shaped curves are generalized and it is shown that every knot
in the minor subfamilies, called sporadic examples, of Berge's lens space surgery is presented 
by a generalized L-shaped curve as a divide knot. 
A formula on the surgery coefficients and the presentation is also generalized.

The Mazur manifold is known as the first example of cork, that is, a contractible 4-manifold 
that can change differential structures of 4-manifolds by cut and reglue with a twisting map. 
The Mazur link is a two-component link that describes the Mazur manifold. Akbulut-Yasui generalized them 
and constructed a sequence of corks. We name their links Akbulut-Yasui links and 
make a complete list of exceptional, i.e., non-hyperbolic integral Dehn surgeries along them. 
We use Martelli-Petronio-Roukema's theorem on exceptional Dehn surgeries along the minimally twisted four chain link.

A framed knot with an integral coefficient determines a simply-connected 4-manifold by 2-handle attachment. 
Its boundary is a 3-manifold obtained by Dehn surgery along the framed knot. For a pair of such Dehn surgeries
along distinct knots whose results are homeomorphic, it is a natural problem: Determine the closed 4-manifold 
obtained by pasting the 4-manifolds along their boundaries. We study pairs of lens space surgeries along
distinct knots whose lens spaces (i.e., the resulting lens spaces of the surgeries) are orientation-preservingly
or -reversingly homeomorphic. In the authors' previous work, we treated with the case both knots are torus knots. 
In the present paper, we focus on the case where one is a torus knot and the other is a Berge's knot TypeVII or VIII,
in a genus one fiber surface. We determine the complete list (set) of such pairs of lens space surgeries and
study the closed 4-manifolds constructed as above. The list consists of six sequences. All framed links and handle
calculus are indexed by integers.

We study lens space surgeries along two different families of 2-component links, denoted by 
$A_{m,n}$ and $B_{p,q}$, related with the rational homology $4$-ball used in J.Park's 
(generalized) rational blow down. We determine which coefficient $r$ of the knotted component
of the link yields a lens space by Dehn surgery. The link $A_{m,n}$ yields a lens space
only by the known surgery with $r=mn$ and unexpectedly with $r=7$ for $(m,n)=(2,3)$.
On the other hand, $B_{p,q}$ yields a lens space by infinitely many $r$. Our main tool for the 
proof is the Reidemeister-Turaev torsions, i.e. Reidemeister torsions with combinatorial Euler 
structures. Our results can be extended to the links whose Alexander polynomials are same with
those of $A_{m,n}$ and $B_{p,q}$.

A framed knot with an integral coefficient defines a simply-connected 4-manifold whose boundary is
the 3-manifold obtained by Dehn surgery along the framed knot, by 2-handle attaching.
For a pair of such surgeries along distinct knots whose results are homeomorphic,
it is a natural problem: Determine the closed 4-manifold obtained by pasting the 4-manifolds along their boundaries.
  We determine the complete list (set) of pairs of integral surgeries
along torus knots whose resulting manifolds are orientation preserving/reversing homeomorphic
lens spaces, and study the 4-dimensional problem above. 
The list consists of five sequences. All framed links and Kirby calculus are indexed by integers.

It is proved that every knot in the major subfamilies of J.~Berge's lens space surgery
(i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped (real) 
plane curve as a {¥it divide knot} defined by N.~A'Campo, in the context of 
the singularity of the complex curves. The corresponding plane curves are constructed.
Such presentations support us to study each knot of lens space surgery itself,
and the relationship among the knots in the set of lens space surgeries.

In the present note, we will show that there are infinitely many torus knots with twists containing essential tori in the exteriors.

A necessary condition on Alexander polynomial ${¥mit ¥Delta}_K(t)$
for lens surgery, i.e., for a knot $K$ to yeild a lens space 
by $p/q$-surgery, is proved.
The condition is an equality as such a product of ${¥mit ¥Delta}_K(t^i)$ 
for some $i$'s equals to $t^m$ for an integer $m$ modulo the ideal $(t^p-1)$.
As examples, the case of pretzel knot of type $(-2,3,7)$ and its some 
extension are demonstrated.

Let $K$ be a knot in a homology $3$-sphere $¥Sigma$. We denote the resulting manifold of $p/q$-surgery along $K$
by $¥Sigma(K;p/q)$.
We study which $¥Sigma(K;p/q)$ is a lens space in the case that 
$K$ has the same Alexander polynomial as that of the $(-2, m, n)$-pretzel knot,
where $m$ and $n$ are positive odd numbers.
First, we point out that $S^3(K;p)$ for such a knot is not a lens space if both $m$ and $n$ are greater than $3$
by P.~Ozsv¥'ath and Z.~Szab¥'o's recent result.
Next, by number-theoretical study on Reidemeister torsion, we prove that $¥Sigma(K;p/q)$ is not a lens space in some cases.
Our method works for knots not only in $S^3$ but also in homology $3$-spheres.
We also extend a certain proposition of R.~Fintushel and R.~Stern on the second parameters of lens space
obtained by Dehn surgery on knots. 

The pretzel knot of type (-2,3,7) is very well known as a 
starting example of the lens surgery (i.e. hyperbolic knots yielding
lens spaces by Dehn surgery), and its Alexander polynomial
also has been studied well in the theory of algebraic integers.
In this paper, the author studies two equalities concerning the 
Alexander polynomials of this knot and he also explains his recent
research concerning this knot.

We give two geometric methods to construct plane curves giving cable knots of torus knots
via A'Campo's divide knot theory, related to 
both singularity theory and knot theory.
We point out a relationship between ¥lq¥lq area¥rq¥rq of the plane curves and coefficients of {¥it finite Dehn surgery},
which is Dehn surgery yielding three-dimensional manifolds with finite fundamental group. 

A family of knots yielding graph manifolds by Dehn surgery is constructed. 
Knots in the family are parametrized (essentially) by four integers.
The proof is by a sequence of Kirby calculus and related to the resolution 
of the singularity of the complex curve of $z^a -w^b=0$.
This is an extension of the author's work ¥cite{Y1} 
on J.~Berge's family of lens surgery.

In 1990, John Berge described several families of knots in the three-dimensional sphere 
which have non-trivial Dehn surgeries yielding lens spaces. 
We study a subfamily of them from the view point of resolution of singularity of complex curves and surfaces,
Kirby calculus in topology of four-dimansional manifolds and A'Campo's divide knot theory. 

We are concerned with plane curves of a certain type $C(p,q,r)$

and their corresponding links $L(C(p,q,r))$ via A'Campo's divide theory,
where $p,q,r$ are positive integers with $1 ¥leq p ¥leq q ¥leq r$.
We will point out that 2-fold covering spaces of the $3$-dimensional sphere $S^3$ branched along $L(C(p,q,r))$
($2$-branched coverings, for short) is represented by Kirby-Melvin's grapes.
We will also refer to some other related topics.