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Useful facts about tangles:
\v


\item{(1)} Both $N(a/b)$ and $D(a/b)$ are 4-plats. The knot/link $N(a/b)$
is
the 4-plat $S(a, -b)$. The knot/link $D(a/b)$ is the 4-plat $S(b, a)$.
\v

\item{(2)} $N((c_1, ..., c_n)) = N((c_n, ..., c_1)) = N((c_1, ..., c_n 
- 1, 1)) $

\rightline{$= N((1, c_1 - 1, ..., c_n))$}

\item{(3)} The tangle corresponding to ${a_1 \over b_1}$ is the same as
the
tangle corresponding to ${a_2 \over b_2}$ if and only if ${a_1 \over b_1}
= {a_2 \over b_2}$.
\v


\item{(4)} The 4-plats $N(a_1/b_1)$ and $N(a_2/b_2)$, $a_i \geq 0$, are
are
the same if and only if $a_1 = a_2$ and $b_1b_2^{\pm 1} \cong 1$ (mod
$a_1$).

i.e. $b_1 = b_2$ (mod $a_1$) or
$b_1 b_2$ = 1 (mod $a_1$) 

or equivalently, 
 $b_1 = b_2$ (mod $a_1$) or
$b_1 = b_2^{-1}$ (mod $a_1$) 

\v
\item{(5)} The mirror image of $N({a \over b})$ is $N(-{a \over b})$

\item{(6)} $N({a \over b})$ is achiral if and only if $b^2 \cong -1$ (mod
$a$) or $2b = 1$ (mod $a$).


\item{(7)} {$N({a_1 \over b_1} + {a_2 \over b_2}) = N({a_1b_2 +
a_2b_1 \over a_1b_2' +a_2'b_1})$ where $a_2'b_2 - a_2b_2' = 1$ [5].
\v

\item{(8)} $N(A + C) = N(C + A)$

\item{(9)} $A + C \not= C + A$


\item{(10)} ${a_1 \over b_1} + {a_2 \over b_2} \not=$ a rational tangle
unless
either $b_1 = \pm 1$ or $b_2 = \pm 1$.
\v



\item{(11)} ${a \over b} + t = {a + bt \over b} $

\v 
\item{(12)} $D(A + C) = D(A) \# D(C)$.

\v
  
\item{(13)} $N({a_1 \over b_1} + ... + {a_n \over b_n})$ is a Montesinos
knot which is not a 4-plat unless at most two of the tangles ${a_i \over
b_i}$ are non-integral.

\v

\item{(14)} N(A + C) = 4-plat implies at least one of A or C is
rational or
locally knotted [8][9].
\v
  
\item{(15)} If A is a prime tangle and C is locally unknotted
and not the $\infty$-tangle, then 
A + C is a prime tangle  [9][10].

}

The tangle, C, is locally unknotted if every 3-ball which intersects the
arcs of C in exactly two points contains an unknotted spanning arc.  A
tangle is locally knotted if it is not locally unknotted.  A
tangle is prime if it is neither rational nor locally knotted.


\item{(16)} $R$ is a rational tangle if and only if its double branch 
cover is a solid torus.

\item{(17)} $M$ a ambient isotopic to a sum of rational tangles if and 
only if its double branch cover is a Seifert fiber space with orbit 
surface a disk and $n > 0$ exceptional fibers [11].


\item{[5]}  C. \spa{Ernst}, D. W.  \spa{Sumners}, {\it  A calculus for
rational tangles:  applications to DNA recombination}\/, Math. Proc.
Camb.
Phil. Soc. 108 (1990), 489--515.


\item{[7]} J. R. Goldman, L. H. Kauffman,  {\it Rational
tangles},
Adv.
in
Appl.
Math. 18 (1997), no. 3, 300--332.

\item{[8]} S. Bleiler, {\it Knots prime on many strings}, Trans. Amer.
Math. Soc. 282 (1984), no. 1, 385--401.



\item{[9]} W. B. R. Lickorish, {\it  Prime knots and tangles},
Trans. Amer. Math. Soc.
267 (1981), no. 1,
321--332.

\item{[10]}  C. \spa{Ernst}, D. W.  \spa{Sumners}, {\it
Solving
tangles equations arising in a DNA recombination model}\/, Math.   
Proc. Camb.
Phil. Soc. 124 (1998).

\item{[11]} C. Ernst,  {\it  Tangle equations}, J. Knot Theory
Ramifications 5 (1996), no. 2,
145--159.


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