Addition and closure of tangles

The following are standard definitions and notations found in many knot theory books.

The addition $T_1+T_2$ of two tangles $T_1$, $T_2$ is another tangles defined from the original two as depicted in Fig. . The closures are two methods of obtaining a link from a tangle by closing the end points, and there are two ways called the numerator $N(T)$ and denominator $D(T)$ of a tangle $T$, defined as depicted in Fig. .

Figure: Addition of tangles

Figure: Closures (numerator $N(T)$, denominator $D(T)$) of tangles

There is a family of ``trivial'' or ``rational'' tangles, some of which are depicted in depicted in Fig. . These are obtained from the trivial tangle of two vertical straight arcs by successively twisting end points vertically and horizontally. See, again, [Mura96] or [Ad94], for example, for more details.

Figure: Some rational tangles