Aug 23, 2018 · Understanding the Hopf Link. Ask Question Asked 6 years, 7 months ago. Modified 6 years, 7 months ago.

Feb 12, 2019 · Title says it all basically. I'm trying to understand why the fundamental group of the hopf link (or really, the compliment of the Hopf link) is abelian. I mean, in a certain way I understand it, b...

One could outlaw the Hopf link on size grounds and add "of at least 3 components" to the definition of a Brunnian link, but then the family of rubberband Brunnian links is missing its first member (actually, its second; but the first is a fancy way of drawing the unlink).

Aug 3, 2022 · This ambiguity also arises when a link is an obvious connect sum with a Hopf link. It's a known problem that Scott Morrison, one of the authors of KnotTheory, once warned me about. For illustration, here are the four ways of interpreting orientations for the PD code [(2,3,1,4), (4,1,3,2)] :

Jul 1, 2015 · $\begingroup$ I assume you're referring here to the Hopf link. Represent the linked circles as the z-axis $\cup \{\infty\}$ and the unit circle in the xy-plane in $\mathbb R^3 \cup \{\infty\}$. Delete these. Show that this deformation retracts onto a …

Moving on to the Hopf link, let's focus on the topmost (negative) crossing. Resolving that one into a positive crossing will leave you with a two-component unlink. Resolving it into a non-crossing leaves you with an unknot.

Oct 26, 2017 · This uses that the HOMFLY polynomial for the split link of two unknots is $\frac{-\ell-\ell^{-1}}{m}$, which either comes from the split link formula or a quick calculation: HOMFLY of the Hopf link with one of the components reversed in orientation gives the same polynomial but with $\ell$ and $\ell^{-1}$ swapped.

Aug 11, 2021 · Hopf fibration from Hopf link. The Hopf fibration is a continuous function from the 3-sphere (a three-dimensional surface in four-dimensional Euclidean space) into the more familiar 2-sphere, with the property that the inverse image of each point on the 2-sphere is a circle.

Mar 18, 2021 · After a small rotation, we can put those rotation axes into the "standard" 2-component Hopf link position: There is another way to visualize this. The 3-components of the link lie on a Clifford torus, and we can stereographically project in …

Feb 2, 2023 · I understand that the linking number of the Hopf link parameterized above is $-1$ by combinatorial considerations -- and therefore the integral in question is $-4\pi$-- but I am curious to see if this integral can be evaluated directly.