Obstructions to tangle embeddings

We use quandle cocycle invariants as obstructions to embedding tangles in knots. To use the cocycle invariants, we first define cocycle invariants for tangles.

Definition 4.1   Let $T$ be a tangle and $X$ a quandle. A (boundary-monochromatic) coloring ${\cal C}: {\cal A} \rightarrow X$ is a map from the set of arcs in a diagram of $T$ to $X$ satisfying the same quandle coloring condition as for knot diagrams at each crossing, such that the (four) boundary points of the tangle diagram receives the same element of $X$.

For a coloring ${\cal C}$ of a tangle diagram $T$, a region colorings are defined in a similar manner as in the knot case. In this case, we allow region colors to change (not necessarily colored by the same element as the one assigned to the boundary points).

Denote by ${\rm Col}_X(T)$ the set of colorings of a diagram of $T$ by $X$. Denote by ${\rm Col}_X(T, s)$ the set of colorings of a diagram of $T$ by $X$ with the color of the leftmost region (between the boundary arcs NW and SW) being $s \in X$. It is seen in a way similar to the knot case that the number of colorings $\vert{\rm Col}_X(T)\vert$ does not depend on a choice of a diagram of $T$, and that the set of colorings are in one-to-one correspondence between Reidemeister moves.

The quandle $2$- and $3$-cocycle invariants are defined for tangles in a manner similar to the knot case, and denoted by $\Phi_{\phi}(T)$.

Definition 4.2   The inclusion of multisets are denoted by $\subset_m$. Specifically, if an element $x$ is repeated $n$ times in a multiset, call $n$ the multiplicity of $x$, then $M \subset N$ for multisets $M$, $N$ means that if $x \in M$, then $x \in N$ and the multiplicity of $x$ in $M$ is less than or equal to the multiplicity of $x$ in $N$.

Theorem 4.3   Let $T$ be a tangle and $X$ a quandle. Suppose $T$ embeds in a link $L$. Then we have the inclusion $\Phi_{\phi}(T) \subset_m \Phi_{\phi}(L)$.

Proof. Suppose a diagram of $T$ embeds in a diagram of $L$. We continue to use $T$ and $L$ for these diagrams. For a coloring ${\cal C}$ of $T$, let $x$ be the color of the boundary points. Then there is a unique coloring ${\cal C}'$ of $L$ such that the restriction of ${\cal C}'$ on $T$ is ${\cal C}$ and all the arcs of $L$ outside of $T$ receive the color $x$. Then the contribution of $\sum_{\tau \in T} B( {\cal C}, \tau )$ to $\Phi_{\phi}(T)$ is equal to the contribution $\sum_{\tau \in L} B( {\cal C}', \tau )$ to $\Phi_{\phi}(L)$, and the theorem follows. $\Box$

In this project we examine the cocycle invariants of tangles in the table and those of knots in the table that do not satisfy the condition of the above theorem, detecting the tangles that do not embed in knots in the tables.