Conway Notation

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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 LinKnots` (by M.Ochiai and N.Imafuji, S. Jablan and R. Sazdanovic). See Using the package LinKnots` for more information.

(For In[1] see Setup)

In[2]:=

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

In[2]:= ?Kauffman Kauffman[K][a, z] computes the Kauffman polynomial of a knot or link K, in the variables a and z.

In[3]:= Kauffman::about The Kauffman program was written by Scott Morrison.

(* to be updated*)

Conway notation was introduced by J.H.~Conway in 1967 (Conway, 1970). ... The main building block is tangle. A {\it tangle} in a knot or link projection is a region in the projectionplane $\Re ^2$ (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 two operations: {\it sum} and {\it product},leading from tangles $a$, $b$ to the new tangles $a+b$, $-a$, $a\,b$, where $-a$ is the image of $a$ in NW-SE mirror line, $a\,b = -a+b$, and $-a = a\,0$. A third operation: {\it ramification} is defined as $a,b = -a-b$.

Pic

A {\it rational tangle} is any tangle obtained from elementary tangles using described operations. A {\it rational knot or link} is a numerator closure of a rational tangle. A knot or link is called {\it algebraic} if it can be obtained as a closure of a tangle obtained from rational tangles using operations product and sum. Knot or links that can not be obtained in this way are called {\it non-algebraic}. Conway notation for non-algebraic knots and links is a bit more complicated. It consists of symbol of basic polyhedron $P^*$ where $P$=$ni$ where $n$ is the number of vertices and $i$ is the index in the list of basic polyhedra with $n$ crossings.A 4-valent graph without digons is called a {\it basic polyhedron}or more precisey it is 4-regular 4-edge-connected, at least 2-vertex connected plane graph. Non-algebraic knots and links can be obtained by substituting tangles in $P^*$ by substituting tangles $t_1$, $\ldots$, $t_k$ in appropriate places is denoted by $P^*t_1\ldots t_k$, where the number of dots between two successive tangles shows the number of omitted substituents of value 1.

For example, the knot 4_1 is denoted by "2 2", knot 9_5 by "5 1 3",  link 5_1^2 is denoted by "2 1 2" , link 9_21^2 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 {\it 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 $9_{15}$ gives projection with 5 not 4 digons.