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Cut Stabilization
A few months ago, I wrote about a generalization of thin position for bridge surfaces of knots and links (introduced by Maggy Tomova [1]) that comes from considering cut disks, i.e. compression disks...
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The generalized Scharlemann-Tomova conjecture
Soon after John Hempel introduced the notion of (curve complex) distance for Heegaard splittings, Kevin Hartshorn showed that the existence of an incompressible surface implies a bound on the distance...
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Dehn filling and genus dropping
A common problem in low-dimensional topology is to ask how the topology and geometry of a manifold changes if you glue a solid torus into one of its torus boundary components (also known as Dehn...
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The reducible automorphism conjecture
Recall that the mapping class group of a Heegaard splitting is the group of automorphisms of the ambient 3-manifold that take the Heegaard surface onto itself, modulo isotopies of that keep on itself....
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The minimal genus Heegaard splitting conjecture
Today, I will continue on my quest to find the most interesting conjectures about Heegaard splittings. (Most of these conjectures, including this one, fail criteria one and two in Daniel’s recent post,...
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Bill Thurston is dead at age 65.
Bill Thurston passed away yesterday at 8pm, succumbing to the cancer that he had been battling for the past two years. I don’t think it’s possible to overstate the revolutionary impact that he had on...
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More than you probably wanted to know about Scharlemann’s no-nesting Lemma
This post is going to be a bit more technical than usual (though not necessarily any more coherent). As I’ve been working on porting thin position techniques to the analysis of large data sets and...
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Morse-Novikov number and tunnel number
Someone recently pointed out to me a paper by A. J. Pajitnov [1] proving a very interesting connection between circular Morse functions and (linear) Morse functions on knot complements. (A similar...
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Topologically minimal surfaces – More common than you might think
Before I get back to train tracks (as I had promised in my last post), I wanted to point out some interesting recent work on topologically minimal surfaces. The definition of topologically minimal...
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The Bridge Spectrum
A knot in a three-manifold is said to be in bridge position with respect to a Heegaard surface if the intersection of with each of the two handlebody components of the complement of is a collection of...
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Update on subadditivity of tunnel number
A few months ago, I wrote a blog post about the interesting phenomenon that the tunnel number of a connect sum of two knots may be anywhere from one more than the sum of the tunnel numbers to a...
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