Mathematics Problem Archive

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...

Characterize those elements of ˆA(S1)conn of the form log Z(K), or those elements of Bconn of the form log⊔ Z(K)....

(J. Roberts) Give a good topological construction of the Kont- sevich integral....

Construct the Kontsevich invariant (i.e. a universal Vassi liev invariant) with coefficients in a finite field....

(D. Bar-Natan, A. Haviv) ι ( Z(O) ) = closure ( exp (1 2 ( − ) ) ), where Z(O) denotes the Kontsevich invariant of the trivial knot (see [3 5]) and ι ...

(M. Polyak) Construct the “Kontsevich invariant” (i.e. a uni- versal finite type invariant) of virtual knots in − →A (I). (See also Conjecture 2.17.)...

(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...

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....

(J. Roberts) What is graph cohomology the cohomology of?...

(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) 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....