Can someone explain Conway Notation for Knots and Links?

Question

I'm trying to wrap my head around the way Conway notation works for knots/links.

In particular I'm not sure of the difference between the notation of the form $[4 2]$ and notation of the form $6^*2:2:20$.

In particular I'm not really sure I understand the process of converting a tangle to a knot/link diagram.

Answer 1

A knot/link diagram, from the point of view of Conway notation, consists of a basic polyhedron, which is actually a 4-regular simple plane graph, and the algebraic tangles inserted into its vertices.

Generally, the notation for a diagram consists of the symbol of the basic polyhedron (say, $1^*$, $6^*$, or $10^{**}$, where the number indicates the number of vertices and the number of stars specifies a particular basic polyhedron from the list) and the list of the symbols of tangles separated by dots; the number of tangles equals to the number of vertices since a tangle should be inserted into every vertex. And yes, in order for this to work, for every basic polyhedron we must specify how exactly tangles are inserted into vertices (Conway shows this with L's in his drawings).

However, Conway developed a shorthand notation, in which some dots are reduced. For example, $6^*2\colon\!2\colon\!2\,0$ expands to $6^*2.1.2.1.2\,0.1$. Moreover, some symbols of basic polyhedra are not written down explicitly. Namely:

• If $t$ is a tangle, then $t$ is also the (shorthand) Conway notation for $1^*t$. To obtain knot/link $t$ from tangle $t$, we join the top endpoints of $t$ together and the bottom endpoints also together. This may be regarded as "converting a tangle to a knot/link diagram". Such knots and links (based on the polyhedron $1^*$) are called algebraic. Non-algebraic knots are built from more than one tangle.
• If there are more than one tangle symbols separated by dots without the basic polyhedron symbol, then it is meant to be $6^*$, so the above example can be written as $2\colon\!2\colon\!2\,0$. If there is also a dot in the beginning, then it is $6^{**}$; however, $6^{**}$ is isomorphic to $6^*$, so it is essentially the same basic polyhedron with alternative way to insert tangles (in order for obtained notation for knots to be nicer).

Until now, we didn't discuss the notation for tangles. Algebraic tangles, which are the only tangles used in the Conway notation, are built from integer tangles by certain set of operation (in particular, $2\,0$ is a certain combination of integer tangles $2$ and $0$; and $20$, twenty, is also a tangle, of course). An integer tangle, in turn, consists of $n$ twists and is denoted as $n$, or $[n]$. The former notation is usually preferred in the discussion of knots and links, not only tangles, because (I presume) otherwise the notation would be overburdened with square brackets.

I hope this helps; if not, please specify what exactly you already know and understand and what needs further discussion.