Dork Scratchings

Short answer: If there were a simple formula to follow there would also be one for proving or disproving any mathematical statement. Before setting out a sketch of a proof I need to be more clear about the problem.  I want to rule out the trivial method of untangling where you simply thread one end backwards through the tangle till it meets the other end.  So we're going to imagine it's a loop of string that's knotted, and "untangling" means getting back to a simple loop, without magically passing one bit through another. For most closed loops it's not possible to untangle to a loop.  For example if you look at the Trefoil Knot above (a.k.a. Granny Knot) it's obvious that no amount of fiddling with it is going to get rid of the knotty bit and leave just a loop.  But if your loop $L$ can be deformed into a simple loop then it follows that the complement in 3D space $\mathbb{R}^3 - L$ can be deformed continuously into the complement of the simple loop $\

  I picked up this fantastic little book in Cambridge market a few days ago.  Published in 1908 by the Society for the Promotion of Christian Knowledge , it summarizes a lecture performed by Professor J Perry in 1890 in Leeds. The theme of the Romance of Science series is that remarkable and unexpected things can be discovered from the study of the apparently mundane.  In this lecture Professor Perry performs a series of experiments on various designs of spinning top and shows i.) What causes the precessional motion of a spinning top ii.) That a subterranean race could determine that we live on a spinning planet without ever seeing the stars iii.) Why the precessional motion of a top is in the same orientation as the spin, but the precessional motion of the Earth is in the opposite orientation iv.) Why some spinning tops stand up when you spin them on their side and even v.) A model of how a magnetic field might cause the observed rotation of plane polarized light The qualitative exp