Mathematics Problem Archive

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

(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 ...

(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....

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 ...

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...

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...

(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 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....

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?...

(R. Fenn, C. Rourke, B. Sanderson) H3(Rp)∼ = Z⊕ Z/pZ for p prime....

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

Is the Burau representation of B4 faithful?...

(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 $\pi_1(M)$....

(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(...