\magnification 1900 \parindent 0pt \parskip 10pt \pageno=1 \hsize 7truein \vsize 9.2truein \def\spa{\hskip 5pt} \def\u{\vskip -10pt} \def\v{\vskip 3pt} \def\w{\vskip 13pt}  Ernst, C.; Sumners, D. W. A calculus for rational tangles: applications to DNA recombination. Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 3, 489--515. Vazquez, Mariel; Sumners, De Witt; Tangle analysis of Gin site-specific recombination. Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 3, 565--582. Darcy, Isabel K. Biological distances on DNA knots and links: applications to XER recombination. Knots in Hellas '98, Vol. 2 (Delphi). J. Knot Theory Ramifications 10 (2001), no. 2, 269--294. math.GT/0501173; Title: Classification of Tangle Solutions for Integrases, A Protein Family that Changes DNA Topology; Authors: Dorothy Buck, Cynthia Verjovsky Marcotte math.GT/0412467; Title: Tangle Solutions for a Family of DNA-Rearranging Proteins; Authors: Dorothy Buck, Cynthia Verjovsky Marcotte Useful facts about tangles: \v p. 491: 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 ${a_1 \over b_1} = {a_2 \over b_2}$ %% \cite{Conway, BZ, Mont19}. \v \eject Parity: \vfill \vfill In order to determine the parity of ${a \over b}$, the Euler bracket function will be used. Let $E[x_1,..., x_n]$ be the Euler bracket function which equals the sum of products of the $x_i$'s where zero or more disjoint pairs of consecutive $x_i$'s are omitted \cite{Roberts}. If $n = 0$ then $E[x_1,..., x_n] = E[] = 1$. If $n < 0$ define $E[x_1,..., x_n] = 0$. Let $[x_n, ..., x_1]$ denote the continued fraction $x_n + {1 \over x_{n-1} + {1 \over ... + { 1 \over x_1}}}$, the fraction corresponding to the tangle $(x_1, ..., x_n)$. The following useful facts for $n \geq 1$ can be found in %%\cite{Roberts}: \hskip 10pt 1.) $E[x_1,..., x_n] = x_1 E[x_2,..., x_n] + E[x_3,..., x_n]$. \hskip 10pt 2.) $[x_n, ..., x_1]$ = $E[x_1, ..., x_n]/E[x_1, ..., x_{n-1}]$. \hskip 10pt 3.) Let $a = E[x_1, ..., x_n], b = E[x_1, ..., x_{n-1}]$. If $y = (-1)^{n+1}E[x_2, ..., x_{n-1}]$ \hskip 27pt and $x = (-1)^{n+1}E[x_2, ..., x_n]$, then $bx - ay = 1$. \eject %%\begin{lemma}{\it Lemma: If $a$ is even, then the tangle ${a \over b}$ has parity 0. If $a$ is odd, then if $b$ is odd, the tangle ${a \over b}$ has parity 1 and if $b$ is even, the tangle ${a \over b}$ has parity $\infty$. %% \label{t-par1} } %%\end{lemma} {\bf Proof.} Induction on n noting that if ${a \over b} = [x_n, ..., x_1]$, then $a = E[x_1, ..., x_n]$ and $b = E[x_1, ..., x_{n-1}]$ \qed\kern0.8pt . 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 p. 493: %% $N((c_1, ..., c_n)) = N((c_n, ..., c_1)) = N((c_1, ..., c_n - 1, 1)) $ %% \rightline{$= N((1, c_1 - 1, ..., c_n))$} 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 $a_1$). %% \cite{Shubert21, BZ, KG}. 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 The mirror image of $N({a \over b})$ is $N(-{a \over b})$ %% $N({a \over b})$ is achiral if and only if $b^2 \cong -1$ (mod $a$) or $2b = 1$ (mod $a$). p. 494: {Lemma 2.1: $N({a_1 \over b_1} + {a_2 \over b_2}) = N({a_1b_2 + a_2b_1 \over a_1b_2' +a_2'b_1})$ where $a_2'b_2 - a_2b_2' = 1$ [5]. \v p. 496: %%\begin{lemma} Thm 2.2: If $N({j \over p} + {f \over g}) = N({a \over b})$, then ${f \over g} = {da - jb' \over pb' - qa}$ for some integers $d$, $q$, and $b'$ such that $pd - qj = 1$, $b'b^{\pm 1} = 1$ mod $a$. %%\label{t-solve} %%\end{lemma} {\bf Proof.} $N({j \over p} + {f \over g}) = N({jg + pf \over dg + qf}) = N({a \over b})$ where $d$ and $q$ are any integers such that $pd - qj = 1$. Since ${f \over g} = {-f \over -g}$ and $N({j \over p} + {-f \over -g}) = N( {-jg - pf \over -dg - qf}) = N({jg + pf \over dg + qf})$, $jg + pf$ will be taken to have the same sign as $a$. Thus $jg + pf = a$ and $dg + qf = b'$ where $b'b^{\pm 1} = 1 ~mod~ a$. Multiplying the first equation by $d$, the second equation by $j$, and solving for $f$ results in $f = (pd - qj)f = da - jb'$. Multiplying the first equation by $q$, the second equation by $p$, and solving for $g$ results in $g = (pd - qj)g = pb' - qa$ \qed\kern0.8pt . Alternate proof use: %%\begin{lemma} {\it $N(A \circ (c_1, ..., c_n) + B) = N(A + B \circ (c_n, ..., c_1))$ if $A$ or $B$ is rational, $n$ odd. } %% \label{t-circle} %%\end{lemma} $N(A + C) = N(C + A)$ $A + C \not= C + A$ %% ${a_1 \over b_1} + {a_2 \over b_2} \not=$ a rational tangle unless either %% $b_1 = \pm 1$ or $b_2 = \pm 1$. \v %% ${a \over b} + t = {a + bt \over b} $ p. 498: Corollary 2.3: Let $B, E$ be distinct rational tangles, Then there are at most two distinct rational solutions to the system of tangle equations $N(X + B) = N({a \over b})$ $N(X + E) = N({z \over v})$ A tangle C is locally unknotted if every 3-ball which intersects the arcs of C in exactly 2 points contains an unknotted spanning arc. A tangle is locally knotted if not locally unknotted. A tangle is prime if it is neither rational nor locally knotted. Lemma 2.4: If $R$ is rational, then $R + R$ is locally unknotted. If $R + R$ is rational, then $R$ is integral. If $R$ is not integral and does not have parity infinity, then $R + R$ is prime \v 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]. $D(A + C) = D(A) \# D(C)$. %% $N({a_1 \over b_1} + ... + {a_n \over b_n})$ 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. p. 499: Lemma 3.1: N(A + C) = 4-plat implies at least one of A or C is rational or locally knotted [8][9]. \v } \th{Cyclic Surgery Theorem [CGLS]:}{ Suppose that M is not a Seifert fibered space. If $\pi_1(M(r))$ and $\pi_1(M(s))$ are cyclic, then $|i(r,s)| \leq1$. Hence there are at most three slopes r such that $\pi_1(M(r))$ is cyclic. } \eject $R$ is a rational tangle if and only if its double branch cover is a solid torus. $M$ 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]. Lemmas 3.1 - 3.7 are used to prove that the double branch cover of $X$ is a torus and hence $X$ is a rational tangle. We will instead use the following to solve equations where $X$ is a generalized M-tangle. This includes the case when $X$ is rational. %%\begin{theorem}(Ernst, \cite{E1}) {\it If $N(U + {f_1 \over g_1}) = N({a \over b})$ and $N(U + {f_2 \over g_2}) = N({z \over v})$ and if $|t| > 1$, then $U$ is a generalized M-tangle or equivalently, $U = A \circ C$ where $A$ is a finite sum of rational tangles and $C$ is a rational tangle. } %%\begin{theorem} {\it $N(U + {0 \over 1}) = N({a \over b})$ 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 $b'x - ay = 1$, %%\begin{equation} \centerline{$N({z \over v}) = N({tb'+ wa \over ty+wx})$} %% \label{e-wnot1} %%\end{equation} 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 %%\begin{equation} \centerline{$N\big({z \over v}\big) = N\big( {tp(pb - qa) \pm a \over tq(pb - qa) \pm b}\big)$} %% \label{e-w1} %%\end{equation} 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 $U$). } %% \label{t-un} %%\end{theorem} %% begin{corollary}{\it 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 $z \mp a$. bottom of p. 502: \vskip -5pt {\bf Theorem 7 [19] } {If $N(U + {f_1 \over g_1}) =$ unknot and $N(U + {f_2 \over g_2}) = N(2z/1)$ where $f_1g_2 - f_2g_1 = \pm 1$, then $U$ is rational.} Theorems 8 and 10 can be used when not all the knots involved are 4-plats and are a generalization of [20] and [21], respectively. \vskip -5pt {\bf Theorem 8 [16]} { If $d_{t/w}(L, N(a/b)) = 1$, then for some integer $n$, and $s \cong w$ mod $t$, $|H_1({\widetilde{L}})| \cong < m, L_1 ~|~ tnL_1 + sm, aL_1 - bnm >$. {\bf Corollary 9}{ If $d_{t/w}(N(a/b), K) = 1$, then there exists an integer $n$ such that $|H_1(M_K)| \equiv |aw + tbn^2|$ (mod $ta$). } {\bf Theorem 10 [22]} { If $d_{t}(K, K') = 1$, then there exist $a \in H_1(M_K)$ and $a' \in H_1(M_{K'})$ such that $\lambda(a, a) \equiv n/|H_1(M_K)|$ and $\lambda(a', a') \equiv m/|H_1(M_K')|$ (mod 1), where $$m = n = { \pm 1 \over t}\big ( |H_1(M_K)| - |H_1(M_K')| \big )$$ or $$-m = n = { \pm 1 \over t}\big ( |H_1(M_K)| + |H_1(M_K')| \big ) $$ } Note that theorem 10 also give information regarding ${t \over kt \pm 1}$ moves since these moves are equivalent to $\pm t$ moves. {\bf Theorem { [5]}}{\it \noindent ~Let $U$ and $R$ be tangles such that $N( U + iR) = $4-plat for some $i \geq 2$, and $N( U + jR) \not= N(U + iR)$ for some j. Then $R$ is a rational tangle. If $i \geq 3$, then $R$ is an integral tangle. } {\bf Proof [10].} If $R$ were locally knotted, then $N( U + iR), i \geq 2$ would be composite. Since 4-plats are prime, $R$ cannot be locally knotted. Suppose $R$ is a prime tangle. By tangle fact (9) $U + (i-1)R$ is rational or locally knotted. By tangle fact (10), $R$ prime implies $(i-1)R$ prime and $U$ must be the $\infty$-tangle or locally knotted. Note that $U$ cannot be the $\infty$-tangle. If $U$ were the infinity tangle, then $N( U + iR) = D(iR) = D(R) \# ... \# D(R)$. Since 4-plats are prime, $D(R) =$ unknot. But $N(U + iR) = D(iR) =$ unknot $= D(jR) = N(U + jR)$, a contradiction. Thus if $U$ is locally unknotted, $R$ must be rational. If $i \geq 3$, then $R$ does not have parity $\infty$ since 4-plats have at most two components. If $i \geq 3$, $U$ is locally unknotted, and $R$ is not integral, then if $U$ is not integral, $U + R$ and $(i-1)R$ are prime by tangle facts (5) and (10). But $N( U + iR) = $4-plat would then contradict tangle fact (9). If $U$ is integral and if $R$ is rational, then $N( U + iR) = $ 4-plat, $i \geq 3$, if and only if $R$ is integral by tangle fact (8). Thus if $U$ is locally unknotted and $i \geq 3$, $R$ must be integral. Suppose $U$ is locally knotted. Then if $U'$ is the tangle formed from $U$ by removing the local knot, then $N( U' + iR) = $unknot, since 4-plats are prime. $N( U + jR) \not= N(U + iR)$ implies that $N( U' + jR) \not= N(U' + iR)$. Since the unknot is a 4-plat and $U'$ is locally unknotted, $R$ is rational if $i \geq 2$ and integral if $i \geq 3$ \,\, \qed\. Cor 6.2: Solve $N(U + P) = N(1/0)$, $N(U + R) = N(-2/1)$, $N(U + R + R) = N((-2, -1, -1)) = N(-5/2)$, \hfil \break $N(U + R + R + R) = N((-1, -1, -1, -1, -1)) = N(-8/5)$ \end \eject \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. \end \vglue 12pt