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Feb
3
comment Hard maths on viXra?
I suppose I would only read a paper on vixra.org if I knew it had a paper that interested me. Similarly, the papers I read on the arXiv I usually hear about some other way -- someone points it out to me, or I see it cited in another paper I'm reading. I don't find it natural to browse paper repositories just to pass time. I suppose I've looked at the arXiv a few times just to get a sense for what some people are up to. That said, I feel like I'm answering a different question than the one you ask.
Feb
2
comment Height function on 2-torus with only 3 critical points
Those are lovely hand-sketches by-the-way. Curtis must have read the Topological Picturebook. Looks almost like Francis drew them.
Feb
2
comment Height function on 2-torus with only 3 critical points
I suppose the first thing you need to do is settle on a notion of complexity. On the analytic end, you could talk about something like the elastic bending energy of the immersion. Perhaps more directly amenable to computation would be the total number of double-points created and destroyed in these level set pictures. I suppose this would be the same as the number of local maxima and minima on the co-dimension two strata of the immersion.
Jan
31
comment Is the digital root of this number ${4^4}^n+1$ always $5$ for all $ n$?
"digital root"?
Jan
27
comment Are there moves between Reidemeister moves?
@JimConant: prime knots have similar problems, but the diagrams tend to be a little more complicated. For example, take a cable of the connect sum above. The above loop can be made to also be a loop for the cable knot, which is prime. The space of the figure-8 knot similarly has non-trivial loops. and on and on...
Jan
27
revised Are there moves between Reidemeister moves?
fix broken image
Jan
22
comment Detection tools for (reduced) suspension
As far as I know, the answer to your question is no.
Jan
19
comment What is modern algebraic topology(homotopy theory) about?
I'm not happy with any of the answers. I think one of the most interesting developments in modern algebraic topology is persistent homology, and the growing applications to data analysis. It has not a particularly large overlap with much of the discussion here. I think this question highlights the diversity of thought that exists in modern algebraic topology. To some, algebraic topology is a beacon of hope for generalists and foundationalists. To others it is something of the opposite: a place where some basic tools have been built that one can use to launch into other fields.
Jan
19
awarded  Nice Answer
Jan
18
comment Is a “knot knot” or “double knot” a thing in knot theory?
Oh, okay. But readers should be aware that your defnition is a special case of the ones you see in the original articles (Schubert) as well as most textbooks and research articles that use the notion.
Jan
18
comment Is a “knot knot” or “double knot” a thing in knot theory?
All the "doubling" type operations you are considering results in prime knots (or links, if you allow yourself to construct objects with more than one component).
Jan
18
comment Is a “knot knot” or “double knot” a thing in knot theory?
All satellite knots are prime.... except for the ones that are not! :) Connect-sums are satellite knots, and they are not prime.
Jan
18
comment Classification of knots by geometrization theorem
There is the variety of "random knot" that comes from using random walks in $\mathbb R^3$. These tend to all have a large number of trefoil summands, more than anything else.
Jan
17
comment Classification of knots by geometrization theorem
In your comment after "edit" I think some qualifiers have been stripped. A satellite knot's alexander polynomial is a product of Alexander polynomials of the companions (suitably re-parametrized) and the "pattern" link. But it's possible that only one of these ingredient Alexander polynomials might be non-trivial. So a satellite knot can still have an irreducible Alexander polynomial. For example, take the connect sum of a knot whose Alexander polynomial is irreducible and a knot that has trivial Alexander polynomial. This is a satellite with irreducible Alexander polynomial.
Jan
17
revised Classification of knots by geometrization theorem
deleted 3 characters in body
Jan
17
revised Classification of knots by geometrization theorem
added 205 characters in body
Jan
17
answered Classification of knots by geometrization theorem
Jan
17
comment What is the Status of Borel conjecture today?
This is a question better answered by 5 minutes in front of a Google prompt.
Jan
13
awarded  Necromancer
Jan
13
answered Algorithm for detecting ribbon or slice links?