Mathematics Problem Archive

Give a collection of fundamental operations on foliations and an explicit family of base foliations such that every tautly foliated manifold $M,\maths...

Suppose $K$ is a non–torus alternating knot. Then essential laminations can be constructed which realize every (nontrivial) boundary slope. Can essent...

It is known that if a $3$–manifold $M$ contains an essential surface of genus $g$, the distance of any Heegaard splitting of $M$ has distance at most ...

Suppose $\mathscr{F}$ is a taut foliation of a hyperbolic $3$–manifold $M$ with two–sided branching. Must there be a leaf $\lambda$ of $\widetilde{\ma...

Suppose $\mathscr{F}$ is a finite depth foliation of a hyperbolic $3$–manifold. What is the relationship (if any) between the Hausdorff dimension of t...

What do short geodesics look like with respect to taut foliations? Is there a universal $\epsilon$ such that for every hyperbolic manifold $M$, every ...

Is there a uniform bound on the Godbillon–Vey invariants of the taut foliations of a hyperbolic manifold in terms of its volume?...

Suppose $\mathscr{F}$ is a taut foliation of a hyperbolic $3$–manifold $M$. Let $$\pi:\widetilde{M} \to L$$ be the projection to the leaf space of $\w...

Suppose $\Lambda$ is an essential lamination of a hyperbolic manifold $M$. Is $\Lambda$ isotopic to a lamination whose curvature is bounded below ever...

What kind of nontrivial ``mapping class elements'' are possible for taut foliations?...

A foliation is taut iff it admits a volume–preserving transverse flow. Pseudo–Anosov flows are good candidates for ``best'' such transverse flows, whe...

Suppose $M$ is atoroidal and $\mathscr{F}$ arises from a slithering over $S^1$. Let $X$ be pseudo–Anosov transverse to $\mathscr{F}$, such that the ti...

If $\mathscr{F}$ is a taut foliation, one can let $\gamma_i$ be a collection of transverse circles to $\mathscr{F}$ intersecting every leaf and study ...

Suppose $\rho:\pi_1(M) \to \mathbb{R}$ is a $1$–cochain with bounded coboundary; i.e. there is a uniform $C$ so that $$|\rho(\alpha) + \rho(\beta) - \...

Does every hyperbolic $3$–manifold admit a taut cone field? That is, a cone field $C$ which is recurrent and supports only homotopically essential loo...

What deformations of a foliation or lamination should be thought of as ``inessential''? For instance –- monotone equivalence, cut–and–shear along a su...

Suppose $\mathscr{F}$ is a minimal taut $C^2$ foliation of an atoroidal $3$–manifold $M$ with $$\mathfrak{gv}(\mathscr{F})[M] \ne 0$$ Is there a choic...

For $\mathscr{F}$ as in the previous question, suppose there is a choice of $\alpha$ for which $\omega$ is a contact form. Is the contact structure de...

Calculate the norm of the fundamental class of a hyperbolic $3$–manifold for some taut foliation $\mathscr{F}$ with two–sided branching....

Let $\mathscr{F},\mathscr{G}$ be taut foliations on a hyperbolic manifold $M$. Are there examples where there is a finite cover of $M$ such that a seq...

Suppose $\mathscr{F}$ is a foliation (possibly $\mathbb{R}$–covered) of a hyperbolic $3$–manifold. Define a foliated Gromov norm using cubical chains....

What kinds of local order structure are there on a family of deformations of a (taut) foliation? Can one use such structures to define co–ordinates on...

What $3$–manifolds admit total taut foliations?...

Are there any interesting examples of total genuine laminations?...

What possibilities are there for (co–oriented) laminations in a $3$–manifold whose transverse spaces are well–ordered? Is there a (useful) theory of ...

([188, Problem 1]) Find a non-trivial knot K with VK (t) = 1....

([188, Problem 2]) Characterize those elements of Z[t, t−1] of the form VK(t)....

(J. Roberts) Why is the Jones polynomial a polynomial?...

(J. Roberts) Define the Jones polynomial intrinsically using homology of local systems....

(J. Roberts) Study the relation between the Jones polynomial and Gromov-Witten theory....

(A. Stoimenow) Does the Jones polynomial V admit only finitely many values of given span? What about the Q polynomia l or the skein, Kauffman polynomia...

(A. Stoimenow) Why are the unit norm complex numbers α for which the value QK (α) has maximal norm statistically concentrated around e11π√ −1/25?...

(M. Kidwell, A. Stoimenow) Let K be a non-trivial knot, and let WK be a Whitehead double of K. Is then degm PWK (l, m) = 2 deg z FK (a, z) + 2?...

(E. Ferrand, A. Stoimenow) Is for any alternating link L, σ(L)≥ min degl ( PL(l, m) ) ≥ min dega ( FL(a−1, z) )?...

(A. Stoimenow) If∇k is the coefficient of zk in the Conway polynomial and c(L) is the crossing number of a link L, is then ⏐ ⏐∇k(L) ⏐ ⏐≤ c(L)k 2k k!?...

(A. Stoimenow) Does min deg a ( FL(a−1, z) ) ≤ 1− χ(L) hold for any link L? If u(K) is the unknotting number of a knot K, does min dega ( FK (a−1, z) ...

Justify the above arguments rigorously....

([220, Problem 1.92 (N)]) Fd(ZK)/Fd+1(ZK) is torsion free for each d....

A(S1; Z) is torsion free....

(T. Stanford) The Dogolazky-Kneissler 2-torsion element in A(↓↓, Z) (see Figure 7) can be embedded into a chord diagram in A(S1, Z) in many ways. Such...

Vassiliev invariants distinguish oriented knots. (See Con jec- ture 3.2 for an equivalent statement of this conjecture.)...

Does there exists a non-trivial oriented knot which can not b e distinguished from the trivial knot by Vassiliev invariant s? (See Problem 3.3 for an ...

(see [220, Problem 1.89 (B)]) For any oriented knot K, no Vassiliev invariants distinguish K from −K. (See Conjecture 3.4 for an equivalent statement ...

(T. Stanford) Can we approximate hG by Vassiliev invariants for other G than dihedral groups?...

(X.-S. Lin [262]) Is the knot signature the limit of a sequence of Vassiliev invariants?...

(N. Okuda [325]) Describe the set {(v2(K) n2, v3(K) n3 ) ∈ R× R ⏐ ⏐ ⏐ K has a knot diagram with n crossings }. (9)...

(S. Willerton [401]) Let v3 be as above. If a knot K has a diagram with n crossings, then |v3(K)|≤ ⌊ n(n2− 1) 24 ⌋....

(T. Stanford) Does Mn have an interesting complementary space in Vn? Consider, for example, the space Nn⊂ Vn of invariants v with the property that v(...

Let I denote an oriented interval. (1) Determine the dimensions of − →A (S1; Q)(d) and− →A (I; Q)(d) for each d. The 6T relation: + + = + + The− → FI ...

(M. Polyak) The following two maps are injective, A(I)(d)−→− →A (I)(d) A(I)(d)/FI−→− →A(I)(d)/− → FI, where they are defined by ↦−→ +....