# Knots from the boundary of Möbius strips

A Möbius strip with one half twist has the unknot as its boundary. One with two half twists has a link of two unknots. One with three half twists has the trefoil knot as its boundary.

Years ago, I remember hearing more about this. Does anyone know of any resources or theorems or anything pertaining to this idea? (Specially which knots/links you obtain from the boundary of a Möbius strip with k half twists?)

• You might be interested in Seifert surfaces, which are surfaces with a given link or knot as a boundary. I know there is an algorithmic way to start with a knot and then produce a surface with that knot as its boundary. – Oiler Apr 10 '16 at 17:00

This answer is going to be less useful than I would wish, because I don't have a reference. But I can at least tell you the answer.

These knots are clearly all elements of the braid group on two strands, called $B_2$. Elements of $B_n$ are generated by taking $n$ separate strands, switching the places of one end of each a pair of strands without switching the other ends,thus half-twisting the two strands together. The group $B_2$ on two strands is particularly simple: it is isomorphic to the integers. We can identify its elements just by saying which integer it corresponds to. This is simply the number $k$ of half-twists of the two strands, which is the same as the number of half-twists of your strip. (Or in the case of negative $k$, half-twists in the other direction.)

If you take a strip and give it $k$ half-twists before gluing the edges, the result is a knot with one component if $k$ is odd, two components if $k$ is even.

When $k$ is odd, the knot is an unknot when $k=1$, a trefoil when $k=3$, a cinquefoil when $k=5$, and so forth. Knot notations usually express the common structure of this family of knots. For example, in Conway notation these are $, , , ,\ldots$. They are all torus knots (meaning they can be embedded in a torus) and in the special torus knot notation they are written $(2, k)$.

When $k$ is even the edge forms two linked circles, linked $\frac k2$ times. (Or in the trivial $k=0$ case two unlinked circles.) For $k=2$ this is the Hopf link, two circles linked in the simplest possible way, and for $k=4$ it is sometimes called Solomon's knot, just like the Hopf link except linked $\left(\frac k2=2\right)$ twice instead of once.

These are also torus knots, again written $(2,k)$ in the special torus knot notation. There is a theorem of torus knots that $(a,b)$ has a single component exactly when $a$ and $b$ are relatively prime, so for your family of knots, there is a single component exactly when $k$ is relatively prime to $2$; that is when $k$ is odd.

It might also be worth pointing out that the knotting of the edges is exactly what determines the behavior when you cut the strip down the middle. Cutting a (single-half-twist) Möbius strip down the middle famously produces a single strip, because the edge is a single unknot. Cutting a double-half-twist strip down the middle produces two linked strips, because the boundary is the Hopf link. Cutting a three-half-twist strip produces a single strip tied in a trefoil knot.

I hope this collection of miscellanea contains something helpful. I suggest you look into the braid groups, because the braid group concept corresponds exactly to what you want to look at: what happens if you take $n=2$ separate strands and cross them exactly $k$ times before joining the ends together.

There is an invariant called the crosscap number, $cc(K)$ which is defined similarily to the genus of a knot, but for non-orientable surfaces. Specifically, for any non-orientable surface $S$ bounded by $K$, we take the minimum of $1-\chi(S)$ over all such surfaces. From the page I linked,

"If a knot is of crosscap number 1, then it bounds a Mobius band, and thus is either a $(2,n)$-torus knot, or has a companion, and hence is not hyperbolic."

I hope this helps. Let me know if there was something else you were looking for.