What are tangles and tangle embeddings?

A tangle is a properly embedded arcs in a ($3$-)ball $B$, also represented by a pair $T=(B, A)$ of arcs $A$ in $B$, but often $T$ is simply regarded as the arcs $A$ if no confusion occurs. In this article a tangle will have four end points unless otherwise specified. A tangle $T$ is embedded in a link (or a knot) $L$ if there is a ball $B$ in $3$-space such that $T=(B, B \cap L)$. Tangles are represented by diagrams in a manner similar to knot diagrams. Usually the end points are located at four corners of a circle at angles $\pi/4$, $3\pi /4$, $5 \pi / 4$ and $7 \pi / 4$, and these end points are labeled by NE, NW, SW, and SE, respectively.

Tangle embeddings have been studied by several authors recently [CL05*,Kre99,KSW00,PSW04*,Rub00]. In this section we prove our main theorem, that uses quandle cocycle invariants for obstructions to tangle embeddings.

In this article we use the table of tangles presented in [KSS03], in particular those with two arcs in a ball. The tangles are parametrized by a pair of numbers in a symbol similar to those representing knots. Those tangles consisting of two arcs are named $5_1$, $6_1 - 6_4$, $7_1 - 7_{18}$ (representing that they listed up to, including, $7$ crossing tangles). Among these, some that are of our interest are depicted in Fig. .

Figure: Some tangles