Is the double-twisted Moebius strip isotopic to the trivial strip? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:22:26Z http://mathoverflow.net/feeds/question/84084 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84084/is-the-double-twisted-moebius-strip-isotopic-to-the-trivial-strip Is the double-twisted Moebius strip isotopic to the trivial strip? Qfwfq 2011-12-22T11:54:49Z 2011-12-22T19:13:20Z <p>Abstractly, on the topological circle $S^1$ there are only two real line bundles, up to isomorphism: the trivial one $\mathcal{O}$ and the Moebius strip $\mathcal{O}(1)$ (thinking of $S^1$ as $\mathbb{RP}^1$). So we have $\mathcal{O}(2k)\cong\mathcal{O}$ and $\mathcal{O}(2k+1)\cong\mathcal{O}(1)$ for any $k$.</p> <p>Let's embed $S^1\hookrightarrow\mathbb{R}^3$ as the set $x^2+y^2=1$ in the $z=0$ plane. We can embed the total spaces $\iota_n:\mathrm{tot}(\mathcal{O}(n))\hookrightarrow\mathbb{R}^3$, respecting the zero section, for example as the union, for $t\in [0,2\pi]$, of the (open) segments joining the two points </p> <p>$((1\pm\frac{1}{2}\cos(nt))\cos(t), (1\pm\frac{1}{2}\sin(nt))\cos(t), \pm\frac{1}{2}\sin(nt))\in\mathbb{R}^3$. </p> <p>Call $\mu(n)\subset\mathbb{R}^3$ the so obtained $n$-twisted Moebius strips.</p> <blockquote> <p>Is $\mu(2k)$ (ambient) isotopic to $\mu(0)$, and $\mu(2k+1)$ to $\mu(1)$?</p> </blockquote> <p>(of course, properly, we're talking about the embeddings $\iota_n$)</p> http://mathoverflow.net/questions/84084/is-the-double-twisted-moebius-strip-isotopic-to-the-trivial-strip/84085#84085 Answer by Ben McKay for Is the double-twisted Moebius strip isotopic to the trivial strip? Ben McKay 2011-12-22T12:01:34Z 2011-12-22T12:01:34Z <p>No. The unit normal bundle to the circle is a torus. Each fiber of the Mobius strip contains a unit vector going in some choice of direction, lying on that torus. That vector travels around the torus representing a homology class, which is clearly not trivial. If we can isotope to the trivial strip, we get a homotopy to the trivial homology class.</p> http://mathoverflow.net/questions/84084/is-the-double-twisted-moebius-strip-isotopic-to-the-trivial-strip/84111#84111 Answer by Adam for Is the double-twisted Moebius strip isotopic to the trivial strip? Adam 2011-12-22T19:13:20Z 2011-12-22T19:13:20Z <p>You could look at the knot type of the boundary of these twisted bands. (Substitute the unit sphere bundle if you want the noncompact version.) For each of your $\mu(k)$ you get a $T(2,k)$ torus link as the boundary. Each of these is non-isotopic. For example, $k=2$ gives the Hopf link and $k=3$ the trefoil. This implies that no two would be ambiently isotopic.</p>