Mathematics Problem Archive

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(...

(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...

(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...

(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....

(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....

(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...

(T. Ohtsuki) Give a “complex structure” to the set of 3- manifolds. More precisely, find an embedding (or, an immersi on) of the set of 3-manifolds to...

(S. Baseilhac, R. Benedetti) Generalize the construction of the QHI for flat principal G-bundles, for Lie groups G different from B. Section 7.4 was wr...