One of my favorite mathematical diagrams: <a href="http://www-math.mit.edu/~andyp/Figures/FIGURE2.pdf" rel="nofollow">http://www-math.mit.edu/~andyp/Figures/FIGURE2.pdf</a>, from Matveev, Fomenko, Algorithmic and computer methods in three-dimensional topology.
I think a lot of human understanding is basically intuition about topological invariants in various "spaces". If you go around asking famous thinkers what they see when they think they all describe similar kinds of imagery, fuzzy shapes that merge and unmerge in various ways as they probe the subject.<p>One good book I've found on the subject is <a href="https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From" rel="nofollow">https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From</a>.
Knots can also be identified with spaces that don't seem "knotty" at first glance.<p>See Thurston's Knots to Narnia[1] and Not Knot[2]. PolyCut[3] is a little applet that can visualize these knotty portals.<p>[1] <a href="https://www.youtube.com/watch?v=IKSrBt2kFD4" rel="nofollow">https://www.youtube.com/watch?v=IKSrBt2kFD4</a>
[2] <a href="https://www.youtube.com/watch?v=zd_HGjH7QZo" rel="nofollow">https://www.youtube.com/watch?v=zd_HGjH7QZo</a>
[3] <a href="http://facstaff.susqu.edu/brakke/polycut/polycut.htm" rel="nofollow">http://facstaff.susqu.edu/brakke/polycut/polycut.htm</a>
hey, I was just reading about Christopher Zeeman [1] who was quite the topologist. He founded the Maths Faculty at the University of Warwick [2].<p>Maths is a truly astonishing discipline begging the question, are we all just made of maths?<p>Zeeman was very quotable: "<i>Technical skill is mastery of complexity while creativity is mastery of simplicity.</i>"<p>[1] Wikipedia entry - <a href="https://en.wikipedia.org/wiki/Christopher_Zeeman" rel="nofollow">https://en.wikipedia.org/wiki/Christopher_Zeeman</a><p>[2] in memory of Zeeman - <a href="http://www2.warwick.ac.uk/knowledge/science/zeeman" rel="nofollow">http://www2.warwick.ac.uk/knowledge/science/zeeman</a>
In the late 1800's knot theory was quite popular with physicists. Now there is the much bigger string theory:
<a href="https://www.sciencedaily.com/releases/2016/02/160210170411.htm" rel="nofollow">https://www.sciencedaily.com/releases/2016/02/160210170411.h...</a><p>On a lighter note I could use some knot theory to explain why earphone or computer cables always seem to tie themselves up, despite my best efforts to keep them apart.
One question if some knot theorists (or at least topologists) are reading along: I can understand why knot theorists are so interested in finding invariants.<p>But now let's define a "dinvariant" ("dual invariant" or "different invariant"): A dinvariant assigns to each knot also some object such that if the knots are <i>different</i> (or topological space are <i>different in their class where they come from</i> (say: are different simplical complexes or different CW complexes), the dinvariant will assign <i>different</i> values. On the other hand, if the knots are equivalent, the assigned values might not be equal.<p>What I want to know is: Why doesn't there seem to exist a theory of dinvariants for knots (or topological spaces)?
Why is it just in 3D? 1D things making knots in 3D seems like it would have immediate analogs for m-dimensional things in n dimensions. Is that not the case? Is it not as rich an area of study or something?
Here's a good video from Numberphile: <a href="https://www.youtube.com/watch?v=aqyyhhnGraw" rel="nofollow">https://www.youtube.com/watch?v=aqyyhhnGraw</a>
Reference: Kawauchi, Akio, and Tomoko Yanagimoto. "What Is Knot Theory? Why Is It In Mathematics?." In Teaching and Learning of Knot Theory in School Mathematics, pp. 1-15. Springer Japan, 2013.<p><a href="http://link.springer.com/chapter/10.1007/978-4-431-54138-7_1" rel="nofollow">http://link.springer.com/chapter/10.1007/978-4-431-54138-7_1</a>
also <a href="https://en.wikipedia.org/wiki/Racks_and_quandles" rel="nofollow">https://en.wikipedia.org/wiki/Racks_and_quandles</a>