Mathematics Problem Archive

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...

Is there some notion of a Godbillon–Vey invariant for a lamination?...

Is there a geometric notion for a $3$–manifold analogous to LERFness for foliations? What properties could a manifold have so that immersed essential ...

Let $\mathscr{F}$ be a taut foliation of $M$. Can leaves of $\mathscr{F}$ be approximated by compact essential surfaces? That is, given a leaf $\lambd...

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

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

What is the weakest useful $2$–dimensional object that might be present in every atoroidal $3$–manifold? For instance, does every hyperbolic $3$–manif...

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

Is there a good notion of taut foliated cobordism? Are there numerical invariants of the equivalence classes this induces on taut foliations which are...

([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)....

Find a 3-dimensional topological interpretation of the Jon es polynomial of links....

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

(J. Roberts) Is there a relationship between values of Jones polynomials at roots of unity and branched cyclic coverings of a knot?...

(J. Roberts) Is there a relationship between the Jones polyno- mial of a knot and the counting of points in varieties defined o ver finite fields?...

(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....

(X.-S. Lin) Describe the set of zeros of the Jones polynomial of all (alternating) knots. -1 -0.5 0.5 1 1.5 -1 -0.5 0.5 1 -1 -0.5 0.5 1 -1 -0.5 0.5 1 ...

(N. Dunfield) Find the relationship between the hyperbolic volume of knot complements and log VK (−1) (resp. log VK(−1)/ log degVK(t)). 3.5 4 4.5 5 5....

Understand Khovanov’s categorification of the Jones polyno - mial....

Categorify other knot polynomials....

(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) ...

(The volume conjecture, [198, 296]) For any knot K, 2π·lim N →∞ log|JN (K)| N = v3||S3− K||, (2) where||·||denotes the simplicial volume and v3 denote...

Justify the above arguments rigorously....

(H. Murakami, J. Murakami, M. Okamoto, T. Takata, Y. Yokota [297]) For a hyperbolic link L, 2π √ −1·lim N →∞ log JN (L) N = CS(S3− L) + √ −1vol(S3− L)...

(H. Murakami) For a torus knot K, calculate CS(S3− K) (giving an appropriate definition of it) and calculate lim log JN (K) N (fixing an appropriate c...

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

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

(X.-S. Lin [262]) Let R be a commutative ring with 1, say Z/2Z. Every weight system A(S1; R)(d)/FI→ R is induced by some Vassiliev invariant RK→ R....

(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 ⌋....