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From Solving unoriented tangle equations involving 4-plats

Theorem 3

$N(U + {0 \over 1}) = N({a \over b})$ and $N(U + {t \over w})
= N({z \over v})$ where $N({a \over b})$ and $N({z \over v})$ are unoriented
4-plats and $U$ is a generalized M-tangle if and only
if the following hold:

(a) If $w \not\cong \pm 1$ mod $t$, then there exists an integer, $b'$ such
that
$b'b^{\pm1} =
1$ mod $a$, and for any integers $x$ and $y$ such that $b'x - ay = 1$,
$$N\big({z \over v}\big) = N\big({tb'+ wa \over ty+wx}\big)$$

In this case, $U = {a \over b'}$ for all $b'$ satisfying the above.

(b) If $w \cong \pm 1$ mod $t$, then there exists relatively prime 
integers, $p$ and $q$, where $p$ may be chosen to be positive,
such that
$$N\big({z \over v}\big) =
 N\big( {tp(pb - qa) \pm a \over tq(pb - qa) \pm b}\big)
$$

In this case, the solutions for $U$ are
  $({da - jb \over pb - qa} + {j \over p})\circ(h, 0)$
  and  $({j \over p} + {da - jb \over pb - qa})\circ(h, 0)$,
 for all $p, q$ satisfying the above, $d$ and $j$ are any integers
such that $~pd - qj = 
1$, and $h = {-w \pm 1 \over t}$ where the $\pm$ sign agrees with
that in (\ref{e-w1}) (note, the choice of $j$ and $d$ such that
$pd - qj = 1$ has no effect on $U$). 
 
{Corollary 1}
 If $|t| > 1$ or if $N({a \over b}) = N({1 \over 0})$, $N({z \over   
v}) =
N({2k \over 1})$, then theorem \ref{t-un} gives all solutions to the   
system of tangle equations: $N(U + {0 \over 1}) = N({a \over b})$ and $N(U
+ {t \over w}) = N({z \over v})$ where $N({a \over b})$ and $N({z \over
v})$ are unoriented 4-plats. 

{Corollary 2}{ Suppose $bx - ay = 1$, $N(U + {0 \over 1}) = N({a
\over b})$ and $N(U + {t \over w}) = N({z \over v})$ where $N({a \over
b})$ and $N({z \over v})$ are unoriented 4-plats.  If $w \not\cong \pm
1$ or if $U$ is rational, then ${t \over w} = {xz - av' \over
bv' - yz - kt}$ and $U = {a \over b + ka}$ OR ${t \over w} = {bz - av'
\over xv' - yz
-
kt}$ and $U = {a \over x + ka}$ where $v'$ is any
integer such that $v'v^{\pm 1} = 1$ mod $z$.  If
$w \cong \pm 1$ mod
$t$, then $t$ divides $z \mp a$.


Theorem 4
A $(0, {t \over w})$ move is equivalent to a $(0, {c
\over d})$ move if and only if ${c \over d} = {t \over w - ht}$ for some
$h$.}

Theorem 5

  Suppose $f_1/g_1 = (c_1, ..., c_n)$, $n$ odd, where $f_1 = E[c_1, ...,
c_n]$ and \break
 $g_1 = E[c_1, ..., c_{n-1}]$.  Let $e_1 = E[c_2, ..., c_n],
i_1 = E[c_2, ..., c_{n-1}]$.
If ${f_2 \over g_2} = (d_1, ..., d_m)$,  let ${t
\over w} = {g_1f_2 - g_2f_1 \over e_1g_2 - i_1f_2} = (d_1,..., d_m -
c_n, -c_{n-1},..., -c_1)$
 and $U' = U \circ (c_n, ..., c_1)$
(or equivalently, if ${t \over w} = (b_1, ..., b_k)$, let ${f_2 \over
g_2} = {te_1 + wf_1 \over ti_1 + wg_1} = (b_1, ..., b_k + c_1, ...,
c_n)$,
 and $U = U' \circ (-c_1, ..., -c_n)$),
 then for any pair of knots $K_1, K_2$,
$$N(U + {f_1 \over g_1}) = K_1 \hskip 0.52in N(U + {f_2 \over g_2}) = K_2$$
\vskip -5pt   
if and only if
\vskip -5pt

$$N(U' + {0 \over 1}) = K_1 \hskip 0.52in N(U' + {t \over w}) = K_2$$

Hence an $({f_1 \over g_1}, {f_2 \over g_2})$ move is equivalent to
a $(0, {g_1f_2 - g_2f_1 \over e_1g_2 - i_1f_2})$ move, and similarly, a
$(0, {t \over w})$ move is equivalent to an $({f_1 \over g_1},  {te_1
+ wf_1 \over ti_1 + wg_1})$.


Moreover if $({f_1 \over g_1}, {f_2 \over g_2})$ is equivalent to $(0, {t \over w})$ then there exists $e_1$ 
and $i_1$ such that $g_1e_1 - f_1i_1 = 1$ an$
{g_1f_2 - g_2f_1 \over e_1g_2 - i_1f_2}$ (or equivalently, ${f_2 \over g_2} = {te_1
+ wf_1 \over ti_1 + wg_1})$.


Theorem 6 ($B = {f_1 \over g_1})

Suppose ${f_1 \over g_1} = (c_1, ..., c_n), n$ odd,
$f_1 = E[c_1, ..., c_{n}], g_1 = E[c_1,..., c_{n-1}]$, $e_1 = E[c_2,
..., c_{n}], i_1 = E[c_2,..., c_{n-1}]$, $t = g_1f_2 - g_2f_1$, $w =
e_1g_2 - i_1f_2$. $N(U + {f_1 \over g_1}) = N({a
\over b})$ and $N(U + {f_2 \over g_2}) = N({z \over v})$ where $N({a \over
b})$ and $N({z \over v})$ are unoriented
4-plats  and $U$ is a generalized M-tangle if and only
if the following hold:

(a) If $w \not\cong \pm 1$ mod $t$, then there exists an integer, $b'$ such
that
$b'b^{\pm1} =
1$ mod $a$, and for any integers $x$ and $y$ such that $b'x - ay = 1$,
$$N\big({z \over v}\big) = N\big({tb'+ wa \over ty+wx}\big)
$$
In this case, $U = {a \over b'} \circ(-c_1, ..., -c_n) = {-f_1b' + e_1a
\over g_1 b' - i_1 a}$ for all $b'$ satisfying the above.


(b) If $w \cong \pm 1$ mod $t$, then there exists relatively prime
integers, $p$ and $q$, where $p$ may be chosen to be positive,
such that
$$N\big({z \over v}\big) =
 N\big( {tp(pb - qa) \pm a \over tq(pb - qa) \pm b}\big)
$$

In this case,
 $U$ = $(U_1 + U_2)\circ(h, -c_1, ..., -c_n))$ and $U$ = $(U_2 +
U_1)\circ(h, -c_1, ..., -c_n)$ where $U_1 = {j \over p}$, $U_2 =
{da - jb \over pb - qa}$ are both solutions for $U$, for all $p, q$
satisfying the above, and $d$ and
$j$ are any integers such that $pd - qj = 1$, and $h = {-w \pm 1
\over t}$ (note, the choice of $j$ such that $pd - qj = 1$ has no
effect on $U$).

If $|f_1g_2 - f_2g_1| > 1$ or if $|f_1g_2 - f_2g_1| = 1$, $N({a \over
b}) = N({1 \over 0})$ and $N({z \over v}) = N({2k \over 1})$, then
the
above list of solutions to the system of equations, $N(U + {f_1 \over 
g_1}) = N({a \over b})$ and $N(U + {f_2
\over g_2}) = N({z \over v})$, is complete.

\end