Conway Notation

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Some generalities about the Conway notation

Conway notation was introduced by J.H. Conway in 1967 (see [Conway]). The main building blocks for the Conway notation are 4-tangles. A 4-tangle in a knot or link projection is a region in the projection plane ${\mathbb {R} }^{2}$ (or on the sphere $S^{3}$) surrounded with a circle such that the projection intersects with the circle exactly four times. The elementary tangles are $0$, $1$ and $-1$.

Pic

Tangles could be combined and modified by one unary operation $a\mapsto -a$ and three binary operations: sum, product and ramification, taking tangles $a$, $b$ to new tangles $a+b$, $a\,b$ and $a,b$. Here $-a$ is the image of $a$ under reflection in the NW-SE mirror line, $a+b$ is obtained by placing $a$ and $b$ side by side with $a$ on the left and $b$ on the right. $a\,b$ is simply $(-a)+b$, and finally, $a,b=(-a)+(-b)$.

Pic

A rational tangle is any tangle obtained from the elementary tangles using only the operation of product. A rational knot or a rational link is the numerator closure of a rational tangle. A knot or link is called algebraic if it can be obtained as the closure of a tangle obtained from rational tangles using the operations above.

Knot or links that can not be obtained in this way are called non-algebraic. They can all be obtained in the following manner: start with a basic polyhedron $P$, a 4-valent graph without digons, with vertices numbered $1$ through $n$. Now substitute tangles $t_{1}$ through $t_{n}$ into these vertices.

The Conway notation for such knots and links consists of the symbol $ni^{\star }$ of a basic polyhedron $P$ where $n$ is the number of vertices and $i$ is the index of $P$ in some fixed list of basic polyhedra with $n$ vertices, followed by the symbols for the tangles $t_{1}$ through $t_{n}$ separated by dots.

For example, the knot 4_1 is denoted by "2 2", knot 9_5 by "5 1 3", the link L5a1 is denoted by "2 1 2" , link L9a24 by "3 1,3,2" (all of them contain spaces between tangles), etc. A sequence of k pluses at the end of Conway symbol is denoted by +k, and the sequence of k minuses by +-k (e.g., knot 10_76 given in Conway notation as 3,3,2++ is denoted by "3,3,2+2", and the link 9_17^3 given in Conway notation as 3,2,2,2-- by "3,2,2,2+-2"). The space is used in the same way in all other symbols. For example, the knot 10_133 is denoted by "2 3,2 1,2+-1", and the knot 10_154 by "(2 1,2) -(2 1,2)" (with spaces).

For the basic polyhedra with N<10 crossings the standard notation is used (.1 , 6*, 8*, 9*, where the symbol for 6* can be ommitted). For example, the knot 10_95 is denoted by ".2 1 0.2.2", and 10_101 by "2 1..2..2". For higher values of N it is used notation in which the first number is the number of crossings, and the next is the ordering number of polyhedron (e.g., 101*, 102*, 103* for N=10 denoting 10*, 10**, 10***, respectively, and 111*, 112*, 113* for N=11 denoting 11*, 11**, 11***, respectively, etc.).

The order of basic polyhedra for N=12 corresponds to their list made by A.Caudron (Caudron A.: Classification des noeuds et des enlancements. Public. Math. d'Orsay 82. Orsay: Univ. Paris Sud, Dept. Math., 1982), so as 121* till 1212* are denoted the basic polyhedra originally titled as 12A-12L. For N>12 the database of basic polyhedra is produced from the list of simple 4-regular 4-edge-connected but not 3-connected plane graphs generated by Brendan McKay using the program "plantri" written by Gunnar Brinkmann and Brendan McKay (http://cs.anu.edu.au/~bdm/plantri/). PolyBase.m is automatically downloaded and it cointains basic polyhedra up to 16 crossings. In order to work with the basic polyhedra up tp 20 vertices, one needs to open an additional database PolyBaseN.m, for N=17 to N=20 (by writing, e.g. <<PolyBase17.m or Needs["PolyBase17.m"] for N=17).

Note: Together with the classical notation, Conway symbols are given in the book Knots and Links by D.~Rolfsen. However if you try to draw some knots or links from their Conway symbols the obtained projection might be non-isomorphic with the one given in Rolfsen, for example knot 9_15 denoted in Conway notation as 2 3 2 2 gives projection with 5, and not 4 digons.

Conway notation and KnotTheory`

Converting Conway notation to other forms known to KnotTheory` (which is a necessary first step in using most of the KnotTheory` functions on knots in Conway notation) requires the package K2K(KNOT 2000) + LinKnot (by M.Ochiai and N.Imafuji, S. Jablan and R. Sazdanovic). See Using the package LinKnot for more information.

(For In[1] see Setup)

In[2]:=

AppendTo[$Path, "/path/to/LinKnots.m"];

In[3]:= ?ConwayNotation ConwayNotation[s] represents the knot or link whose Conway notation is the string s. ConwayNotation[K], where K is a knot or a link with up to 12 crossings, returns ConwayNotation[s], where s is a string containing the Conway notation of K.

In[4]:= ConwayNotation::about The program ConwayNotation relies on code from the LinKnot package by Slavik Jablan and Radmila Sazdanovic.

[Conway] ^  J. H. Conway, An Enumeration of Knots and Links, and Some of Their Algebraic Properties. In Computation Problems in Abstract Algebra (Ed. J. Leech). Oxford, England: Pergamon Press, pp. 329-358, 1967.