David Austin/文

## 2.有理缠结上的操作

●加法：缠结$T+S$是

●乘法：缠结$T∗S$

$$\frac { 1 } { \left[ a _ { k } \right] } * \left( \cdots * \left( \left[ b _ { 3 } \right] + \left( \frac { 1 } { \left[ a _ { 1 } \right] } * \left( \left[ b _ { 1 } \right] + \frac { 1 } { \left[ a _ { 0 } \right] } + \left[ b _ { 2 } \right] \right) * \frac { 1 } { \left[ a _ { 2 } \right] } \right) + \left[ b _ { 4 } \right] \right) * \ldots \right) * \frac { 1 } { \left[ a _ { k + 1 } \right] }.$$

## 3.翻转和折叠

$$T ^ { h } \sim T \text { and } T ^ { v } \sim T.$$

$$\begin{array} { l } { [ \pm 1 ] + T \sim T + [ \pm 1 ] } \\ { [ \pm 1 ] * T \sim T * [ \pm 1 ] } \end{array} .$$

$$[ 5 ] + \left( \frac { 1 } { [ 2 ] } * \left( [ 2 ] + \frac { 1 } { [ 3 ] } + [ - 2 ] \right) * \frac { 1 } { [ 3 ] } \right) \sim [ 5 ] + \frac { 1 } { [ 8 ] }.$$

$$\begin{array} { l } { A : \boldsymbol { T } \mapsto T + [ 1 ] } \\ { R : \boldsymbol { T } \mapsto T ^ { r} } \end{array}.$$

## 4.缠结色彩和有理数

$$\left[ \begin{array} { l l } { a } & { b } \\ { c } & { d } \end{array} \right]$$

$$\left[ \begin{array} { l l } { 3 }&{ 10 } \\ { 0 }&{ 7 } \end{array} \right].$$

$$\left[ \begin{array} { l l } { a }&{ b } \\ { c }&{ d } \end{array} \right],$$

$$\left[ \begin{array} { l l } { m a + n }&{ m b + n } \\ { m c + n }&{ m d + n } \end{array} \right].$$

$$\left[ \begin{array} { c c } { a }&{ 2 b - d } \\ { c }&{ b } \end{array} \right].$$

$T^{r}$的颜色矩阵是通过旋转$T$的颜色矩阵中的元素得到的:

$$\left[ \begin{array} { l l } { b }&{ d } \\ { a }&{ c } \end{array} \right].$$

\begin{aligned} f ( T + [ 1 ] ) &= f ( T ) + 1 \\ f \left( T ^ { r } \right) &= - \frac { 1 } { f ( T ) } \end{aligned}.

## 6.总结

$$T \sim \left[ a _ { 1 } \right] + \frac { 1 } { \left[ a _ { 2 } \right] + \ldots \frac { 1 } { \left[ a _ { n - 1 } \right] + \frac { 1 } { \left[ a _ { n } \right] } } }$$

## 参考文献

1. John Conway. An Enumeration of Knots and Links，and Some of Their Algebraic Properties. Computational Problems in Abstract Algebra. Oxford，329-358. 1970.

2. John Conway. Tangles，Bangles and Knots with John Conway. UC Berkeley Graduate Lectures. 2014

3. Louis Kauffman and Sofia Lambropoulou. Classifying and Applying Rational Knots and Rational Tangles，Contemporary Mathematics 304，223-259. 2002.

4. Jay Goldman and Louis Kauffman. Rational Tangles，Advances in Applied Mathematics 18，300-332. 1997.

5. Tom Davis. Conway's Rational Tangles.

6. Davis gives some ideas for using Conway's square dance with a group of middle or high school students.

7. DeWitt Sumners. Using Topology to Probe the Hidden Action of Enzymes. Notices of the American Mathematical Society 42 (5)，528-537. 1995.

8. C. Ernst and D.W. Sumners. A calculus for rational tangles: Applications to DNA recombination，Mathematical Proceedings of the Cambridge Philosophical Society 108，489-515. 1990.

 原文链接: http://www.ams.org/publicoutreach/feature-column/fc-2017-08 作者: David Austin，Grand Valley State University 翻译: 程晓亮，吉林师范大学数学学院副教授 邬昊然，吉林师范大学基础数学专业研究生二年级