367 citations · 1.6k across the 63 of their papers we have counts for
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Odd Vassiliev invariants vanish at four loops
Greg Kuperberg
The odd Vassiliev conjecture asserts that all Jacobi diagrams with an odd number of legs vanish modulo the AS and IHX relations. Equivalently, the conjecture asserts that Vassiliev…
Coloring invariants of knots and links are often intractable
Greg Kuperberg, Eric Samperton
Let be a nonabelian, simple group with a nontrivial conjugacy class . Let be a diagram of an oriented knot in , thought of as computational input. We sh…
On TQFT representations of mapping class groups with boundary
Greg Kuperberg, Shuang Ming
We study the TQFT mapping class group representations for surfaces with boundary associated with the gauge group, or equivalently the quantum group $U_q(\Sl(2))$. We show t…
Computational complexity and 3-manifolds and zombies
Greg Kuperberg, Eric Samperton
We show the problem of counting homomorphisms from the fundamental group of a homology -sphere to a finite, non-abelian simple group is #P-complete, in the case that …
Identifying lens spaces in polynomial time
Greg Kuperberg
We show that if a closed, oriented 3-manifold M is promised to be homeomorphic to a lens space L(n,k) with n and k unknown, then we can compute both n and k in polynomial time in t…
Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization
Greg Kuperberg
In this paper we prove two results, one semi-historical and the other new. The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem imp…
Knottedness is in NP, modulo GRH
Greg Kuperberg
Given a tame knot K presented in the form of a knot diagram, we show that the problem of determining whether K is knotted is in the complexity class NP, assuming the generalized Ri…
What is a virtual link?
Greg Kuperberg
Several authors have recently studied virtual knots and links because they admit invariants arising from R-matrices. We prove that every virtual link is uniquely represented by a l…
The Second Hull of a Knotted Curve
Jason Cantarella, Greg Kuperberg, Rob Kusner +1
The convex hull of a set K in space consists of points which are, in a certain sense, "surrounded" by K. When K is a closed curve, we define its higher hulls, consisting of points…
Perturbative 3-manifold invariants by cut-and-paste topology
Greg Kuperberg, Dylan P. Thurston
We give a purely topological definition of the perturbative quantum invariants of links and 3-manifolds associated with Chern-Simons field theory. Our definition is as close as pos…
Quadrisecants of knots and links
Greg Kuperberg
We show that every non-trivial tame knot or link in R^3 has a quadrisecant, i.e. four collinear points. The quadrisecant must be topologically non-trivial in a precise sense. As an…