Mathematics Problem Archive

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

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

(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?...

(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....

Categorify other knot polynomials....

(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...

(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)...

(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....

Determine the dimension of the space of primitive Vassiliev invariants of each degree d. Equivalently, determine the dimension of the space A(S1; Q)(d...

(M. Polyak) Milnor’s µ -invariants of string links can be de- fined similarly as above (see [329]). Find a topological pres entation of a µ - invarian...

(M. Polyak) Establish the Goussarov-Habiro theory for vir- tual knots....

Z(K) = Z(−K) for any oriented knot K, where −K de- notes K with the opposite orientation. (See Conjecture 2.7 for an eq uivalent statement of this con...

Find a combinatorial direct presentation of an associator f or all degrees, in particular, an associator with rational coe fficients....

(J. Roberts) Construct a rational Drinfel’d associator in the context of rational homotopy theory....

(T. Kohno) Construct explicitly a universal invariant of finite type for links in Σ × [0, 1] with values in AΣ. In the case of genus 0 the above probl...

(T. Kohno) Give a deformation quantization of the Poisson algebraAΣ which descends to a deformation quantization of C(MG(Σ)). The above problem will g...

(T. Kohno) Clarify the relation between a deformation quan- tization of C(MG(Σ)) at a special parameter and the space of conformal blocks in WZW model...

(T. Kohno) Determine the image and the kernel of the above map τ. The space of conformal blocks in WZW model is defined as the spa ce of coin- variant...

(T. Kohno) Compute the holonomy of the space of conformal blocks of the twisted WZW model. In particular, determine th e action of the braid group of ...

Classify the isomorphism classes of connected quandles of o rder n for each positive integer n. See Table 4 for a list of connected quandles of order ...

Compute H Q 2 (X) for each connected quandle X. More gener- ally, find a convenient methodology to compute quandle (co)h omology groups. See Table 5 f...

Compute the quandle cocycle invariant Φ α(K) of each knot K for a second cohomology class α of a connected quandle....

Find relations between quandle cocycle invariants and knot in- variants known so far, such as quantum invariants....

([188, Problem 3]) Is the representation of the braid group inside the Temperley-Lieb algebra faithful?...

(The perturbative expansion conjecture) The asymptotic expansion of Z G k (M ) of a closed oriented 3-manifold M is given by Z G k (M ) ∼ k→∞ e−π√ −1(...

(The asymptotic expansion conjecture, J.E. Andersen [6]) Let{c0 = 0, c1,···, cm} be the set of values of the Chern-Simons functional of flat G connect...

(The growth rate conjecture) Let d = max{d0,..., dn}. Then|Z G r (M )| = O(rd). It is well known that the quantum invariants only grows like r to some...

There is a construct of the right measure, say τM (A)1/2 for A∈M i, from the square root of the Reidemeister torsion generaliz ing the non-degenerate ...

(H. Murakami [294]) For any closed 3-manifold M, 2π √ −1·o-lim N →∞ log τ SU (2) N (M ) N = CS(M ) + √ −1vol(M ), where vol(M ) and CS(M ) denote the ...

(H. Murakami) Calculate o- lim log τ SU (2) N (M ) N for Seifert fibered 3-manifolds M....

(S. Morita [228]) Define the Chern-Simons invariant CS(M ) as a topological invariant of any closed oriented 3-manifol d M, and of any knot (link) com...

(J. Roberts) Explain the appearance of modular forms in the Witten invariants....

(K. Habiro, T. Le) For each g as above, there is a (unique) invariant I g(M )∈ R1 of an integral homology 3-sphere M such that for each root of unity ...

(T. Kerler) [Cyclotomic integer TQFT’s] (1) Find explicit/computable bases for the Vp(Σ g) as free modules over Z[ζp]. (2) Show that Vp can be extende...

(Y. Kawahigashi) Suppose we have a semisimple ribbon cat- egory C with finitely many isomorphism classes of simple objects. If the S - matrix is inver...

(Y. Kawahigashi) Suppose we have a semisimple ribbon cat- egory C1 with finitely many isomorphism classes of simple objects, bu t the S -matrix is not...

(Y. Kawahigashi) There are some fusion rule algebras with 6j -symbols that do not seem to arise from quantum groups in [14] and more conjectured candi...

(N. Sato) Construct a well-defined state sum type invariant from a strongly amenable subfactor. Note that, unlike the Ponzano-Regge model, we do not h...

Can the Casson invariant of an integral homology 3-sphere M be characterized by the signature of a certain 4-manifold bo unded by M?...

(C. Lescop) Are the Cappell-Lee-Miller Casson-type SU (n)- invariants of finite type? If so, what are their degrees and th eir weight systems?...

Determine the dimension of the space of primitive finite type invariants of integral homology 3-spheres of each degree d. Equivalently, deter- mine th...

Recognising the Kishino Knot: There have been invented many ways to recognize the Kishino virtual knot (from the unknot): The $3$–strand Jones polynom...

Flat Virtuals: Flat virtual knots, also known as virtual strings , are difficult to classify. Find new combinatorial invariants of flat virtual knots....

Virtuals and the Theory of Doodles: Compare flat theories of virtual knots with theories of doodles. A doodle is represented by a flat diagram in the ...

Welded Knots: We would like to understand welded knots . It is well known that if we admit forbidden moves to the virtual link diagrams, each virtual ...