Mathematics Problem Archive

(D. Thurston) Construct a series of configuration space inte- grals whose value is in − →A(I) so that it gives all finite type invariants of virtual k...

(M. Polyak) Find another way to kill the hidden strata, so that the above three approaches can naturally present the ma pping degree of the same map. ...

(C. Lescop) Is the Kontsevich integral of a (zero-framed) knot equal to the Chern-Simons series of configuration space integrals of the same knot (wit...

(R. Bott) Give a geometric construction of these homology classes coming from Lie algebras. The third and currently best interpretation of graph cohom...

Find a topological construction of the 2-loop polynomial P θ K....

(A. Kricker) Let KT be the knot obtained from a tangle T as shown in Figure 12. Find a presentation of the 2-loop polyn omial P θ KT of KT by using th...

Find a topological construction of the polynomial P ′ K given above. = = = = a + b Figure 13: The multi-linear relations. Here, f (t), g(t)∈ S, and a,...

Find a topological construction of the loop-degree l part of the rational Z invariant Z rat(K)∈A Q[t±1,1/∆ K (t)](∅; Q) of a knot K, for each l....

Find a basis of the space AQ[t±1,1/A(t)](∅; Q)(loop l), for each l, where A(t) is a polynomial with A(1) = 1 and A(t) = A(t−1). In particular, find a ...

[357, 139] The map (29) is injective. In particular, the map (30) is injective....

(T. Kohno) Let Pn(Σ) denote the pure braid group of Σ with n strings. Does there exist an injective multiplicative homo morphism θ: Pn(Σ) →A n(Σ) defi...

Calculate S2,∞(M ) for each oriented 3-manifold M. Find a convenient methodology to calculate it....

(J. Przytycki) Incompressible tori and 2-spheres in M yield torsion in S2,∞(M ) [339]. It is a question of fundamental importance whether other surfac...

If every closed incompressible surface in M is parallel to ∂M, then S2,∞(M ) is torsion free....

(J. Przytycki) Compute S2,∞(F0,3× S1)....

Let F be a surface and I an interval. Describe the algebra S2,∞(F× I)....

Calculate the skein homology based on the Kauffman bracket skein relation....

We define the sl3 skein module Ssl3(M ) of an oriented 3- manifold M by the defining relations of the sl3 linear skein [233, 323]. Calculate Ssl3(M ) ...

Calculate S3(M ) for each oriented 3-manifold M. Find a con- venient methodology to calculate it....

Let F be a surface and I an interval. Describe the algebra S3(F× I)....

Calculate S3,∞(M ) for each oriented 3-manifold M. Find a convenient methodology to calculate it....

Calculate the higher skein modules based on the Kauffman skein relation W 3,∞ i (M ) and ˆW 3,∞(M ) (see below for their definitions)....

Construct invariants of 3-manifolds via a linear skein theo ry based on the Kauffman skein module....

Calculate HS q(M ) for each 3-manifold M....

(J. Przytycki) (i) Find generators of S4,∞(S3, R). (ii) For which parameters of the (4,∞) skein and framing relations, trivial links are linearly inde...

(J. Przytycki, see [286]) (1) There is a polynomial invariant of unoriented links, P1(L)∈ Z[x, t] which satisfies: (i) Initial conditions: P1(Tn) = tn...

(J. Przytycki) For which coefficients of the (4,∞) skein rela- tion is the number of Fox 7-colorings measured by the (4,∞) skein module? Figure 16...

Calculate Wk(M ) for each 3-manifold M....

Define a skein module of 3-manifolds, and calculate it....

Describe (the number of) representations of a knot quandle t o a fixed connected quandle of finite order, say, by using knot in variants known so far,...

Let hX be as above. Then, log hX is not a Vassiliev invariant, unless it is constant. 5.3 (Co)homology of quandles Second cohomology classes of a quan...

(J.S. Carter) Compute H Q i (Sm n ) of Sm n which denotes the quandle of the nth symmetric group with the binary operation given by x∗ y = y−mxym. 5.4...

Compute H 2 Q(X; A) for each X -module A....

Let the notation be as above. Then, extending the definition o f the quandle cocycle invariant, define a knot invariant assoc iated with α, which is, ...

(M. Polyak) Define a quantum quandle....

(C. Rourke, B. Sanderson) Is there a natural quandle space whose cohomology groups are the quandle cohomology groups?...

(S.J. Bigelow) Is the action of B6 on V 6 2 faithful?...

(S.J. Bigelow) Generalise Lawrence’s construction to obtain the irreducible representations of the BMW algebra....

(S.J. Bigelow) Find a larger family of irreducible representa- tions of Bn which includes those coming from the BMW algebra....

Classify all irreducible representations of Bn....

(S.J. Bigelow) Is there a faithful representation of Bn into a group of matrices over ¯Q?...

(see [220, Problem 3.108]) Does there exist a closed 3-manifold M, other than S3, such that τ SO(3) r (M ) = τ SO(3) r (S3) for all odd r≥ 3?...

(S.K. Hansen, T. Takata) Find pairs of non-homeomorphic rational homology 3-spheres that can be distinguished by th eir quantum G invariants τ G r or ...

(S.K. Hansen, T. Takata) Do the family of quantum G invari- ants τ G r or the family of quantum P G invariants τ P G r, G running through all simply c...

Find a 3-dimensional topological interpretation of quantu m in- variants of 3-manifolds....

For non-vanishing $\tau_r^G(M)$, the absolute value $|\tau_r^G(M)|$ depends only on the fundamental group $\pi_1(M)$....

(J.E. Andersen) If such an expansion exists, understand how it is related to the expansion of Ohtsuki and the expansion of Habiro. It will of course b...

(Topological interpretations of the dj ’s) Let Mj be the union of components of the moduli space of flat connections M which has Chern-Simons value cj...

(D. Thurston) Find a series of invariants of a 3-manifold (de- pending on roots of unity) that grows as its hyperbolic volum e (or its simplicial volu...

(D. Thurston) Find a correct generalization of the volume conjecture to other non-compact Lie groups....