Kondo and Sakai independently gave a characterization of Alexander polynomials for knots which are transformed into the trivial knot by a single crossing change. The first author gave a ...characterization of Alexander polynomials for knots which are transformed into the trefoil knot (and into the figure-eight knot) by a single crossing change. In this note, we will give a characterization of Alexander polynomials for knots which are transformed into the 10
132 knot (and into the
(
5
,
2
)
-torus knot) by a single crossing change. Moreover, this method can be applied for knots with monic Alexander polynomials.
A central quantity for the calculation of Alexander polynomial of knots is to use Braids presentations of the given knots. For this purpose, we improved a computer program which is writting in Delphi ...programming language. The program calculates Alexander polynomials of the given knot using free derivative that is obtained from Braids presentation of the given knot.
In this paper we apply computer algebra (MAPLE) techniques to calculate Alexander polynomial of (3,
k)-Torus knots. For this purpose, a computer program was developed. When a positive integer
k is ...given, the program calculates Alexander polynomials of (3,
k)-Torus knots from Alexander matrix.
We show that there exists a one-to-one correspondence between the class of certain block tridiagonal matrices with the entries -1, 0, or 1 and the free monoid generated by 2n generators \sigma _{1}, ...\cdots ,\sigma _{n}, \sigma _{1}^{-1},\cdots , \sigma _{n}^{-1} and relation \sigma _{i}^{\pm 1}\sigma _{j}^{\pm 1} = \sigma _{j}^{\pm 1}\sigma _{i}^{\pm 1}~ (|i-j| \geq 2) and give some applications for braids. In particular, we give new formulation of the reduced Alexander matrices for closed braids.
Torsion-Groups of Abelian Coverings of Links Mayberry, John P.; Murasugi, Kunio
Transactions of the American Mathematical Society,
1982, Letnik:
271, Številka:
1
Journal Article
Recenzirano
Odprti dostop
If $M$ is an abelian branched covering of $S^3$ along a link $L$, the order of $H_1(M)$ can be expressed in terms of (i) the Alexander polynomials of $L$ and of its sublinks, and (ii) a "redundancy" ...function characteristic of the monodromy-group. In 1954, the first author thus generalized a result of Fox (for $L$ a knot, in which case the monodromy -group is cyclic and the redundancy trivial); we now prove earlier conjectures and give a simple interpretation of the redundancy. Cyclic coverings of links are discussed as simple special cases. We also prove that the Poincaré conjecture is valid for the above-specified family of 3-manifolds $M$. We state related results for unbranched coverings.
Let
L
be an alternating two-component link with Alexander polynomial
Δ
(
x
,
y
)
. Then the polynomials
(
1
−
x
)
Δ
(
x
,
y
)
and
(
1
−
y
)
Δ
(
x
,
y
)
are alternating. That is,
(
1
−
y
)
Δ
(
x
,
y
)
...can be written as
∑
i
,
j
c
i
j
x
i
y
j
in such a way that
(
−
1
)
i
+
j
c
i
j
≥
0
.
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