Pillowcases

As was discussed in an older news post, there exists a double-covering of the boundary of a torus onto the boundary of a ball – essentially a way for us to translate things on the surface of a ball to what they would be on the surface of a torus. As we are already well aware, the rational tangles we have been discussing are considered to exists as non-intersecting line segments contained within an open ball. Suppose then that we take those line segments and push them onto the boundary of our open ball. Due to the fact that the torus is a double-covering of the boundary of our open ball containing our tangle then, we now have a way to picture our tangle as it would exist on a torus.

If we “cut” our torus on the disk where all the endpoints of our tangle lie, we get a “tangle in a tube” (for your convenience!). If we grab the endpoints on each side of the tube and and “pull” we flatten the tube into a “pillowcase” like shape with the tangle wrapping all around it, with the endpoints of the tangle at the corners.

Let’s draw all our tangle pillowcases with just one side facing us. Intuitively, solid lines indicate that the tangle is on the side we are looking at, and dotted lines indicate that it is on the opposite side. We draw tic marks on our diagram wherever our tangle wraps around from one side to the other. Note that these tic marks indicate that our solid lines have a “slope” of one. If we note now many sections the tic marks divide the bottom and side of our pillowcase into, however, we can get the slope of the tangle as $\frac{b}{s}$ where $b$ is the number of sections on the bottom of the pillowcase, and $s$ is the number of sections on the side.

Aside from being a neat way to diagram tangles and find their slopes, pillowcases offer us a visual, intuitive look into the rational structure on rational tangles, specifically the operations of addition and multiplication of tangles.

ADDITION

We’ve mentioned earlier about adding tangles by attaching them side-by-side or top-to-bottom by their endpoints. On pillowcases the process is remarkably similar – simply attach the pillowcase diagrams at the corners. Note the way solid and dotted lines connect. As on the the pillowcases before adding them together, they are considered on a certain side of the pillowcase. Thus solid lines on different pillowcases are sort of “redrawn” to be the same line on the added pillowcases. Normally this would seem a violation of our vow not to cut the tangles when we add, multiply, or otherwise rearrange them. In reality though, we are merely stretching the line segment to suit our needs.

Ideally the lines on the edges where we add our pillowcases match up, but this is not always the case. Our method of finding slope from a pillowcase diagram fails if we do not have parallel lines travelling from one side all the way to the other, as we will have differing number of sections on parallel sides of our pillowcase. To remedy this, we have to draw more lines in our pillowcase, making sure to keep things from intersecting, and that the ratio of sections on the side to the bottom remains the same (i.e. that we do not change the slope of the tangles we are adding). It would seem that by doing this we are once again violating the rules we had established to manipulate our tangles while preserving their properties. However, all this process does is insert a linked component into the tangle diagram that can easily be removed by a series of type 2 and 3 Reidemeister moves, meaning we haven’t really changed the tangle at all!

The slope of the resulting tangle is equivalent to sum of the slopes of the two tangles we added together. This process is pleasingly similar to converting rational numbers to have common denominators for addition.

MULTIPLICATION

Ozawa’s paper defines the multiplication of rational tangles $T_1$ and $T_2$ by replacing each crossing of $T_1$ with a copy of $T_2$ if the crossing is positive, or $-T_2$ if the crossing is negative.

On the pillowcase, the process is rather simple. We divide our pillowcase of $T_1$ into a grid based on our tic marks. In each box on the grid, we replace the existing lines with a copy of $T_2$ .

The slope of the resulting tangle is equivalent to the product of the slopes of $T_1$ and $T_2$ . It’s a rather intuitive fact based on the process by which we construct the product tangle. Suppose we have the slope of $T_1$ equivalent to $\frac{a}{b}$ and the slope of $T_2$ equivalent to $\latex frac{c}{d}$. On our grid of $T_1$ , we have $a$ squares along the bottom. Replacing each one of these is a copy of the pillowcase of $T_2$ which each have a bottom divided into $c$ sections. Thus in our product, the bottom of the pillowcase is divided into $a \times c$ sections. By similar reasoning, we determine that the side is divided into $b \times d$ sections. Thus the slope of the product tangle is equivalent to $\frac{a \times c}{b \times d}$ , the product of the slopes of $T_1$ and $T_2$ .

CONCLUSION
As in the past, we let $\phi (T)$ be the map that takes our tangle $T$ to the rational number equivalent to its slope. As we’ve shown with our pillowcases, $\phi$ demonstrates properties of a homomorphism, namely:
$\phi (T_1 + T_2) = \phi (T_1) + \phi (T_2)$ and
$\phi (T_1 \times T_2) = \phi(T_1) \times \phi(T_2)$ .

This opens up possibilities for future research. Most immediately, consider the set of all rational tangles to be the set $A$ , and $Ker(\phi) = \{T|\phi(T) = 0\}$ . If we look at the subgroup of $A$ mod $Ker(\phi)$ , then by the First Isomorphism Theorem, there must exist an isomorphism $\psi: A/Ker(\phi) \rightarrow \mathbb{Q}$ !