Mathematics Problem Archive

(see [220, Problem 1.89 (B)]) For any oriented knot K, no Vassiliev invariants distinguish K from −K. (See Conjecture 3.4 for an equivalent statement ...

(T. Stanford) Can we approximate hG by Vassiliev invariants for other G than dihedral groups?...

(X.-S. Lin [262]) Is the knot signature the limit of a sequence of Vassiliev invariants?...

(N. Okuda [325]) Describe the set {(v2(K) n2, v3(K) n3 ) ∈ R× R ⏐ ⏐ ⏐ K has a knot diagram with n crossings }. (9)...

(S. Willerton [401]) Let v3 be as above. If a knot K has a diagram with n crossings, then |v3(K)|≤ ⌊ n(n2− 1) 24 ⌋....

(T. Stanford) Does Mn have an interesting complementary space in Vn? Consider, for example, the space Nn⊂ Vn of invariants v with the property that v(...

Let I denote an oriented interval. (1) Determine the dimensions of − →A (S1; Q)(d) and− →A (I; Q)(d) for each d. The 6T relation: + + = + + The− → FI ...

(M. Polyak) The following two maps are injective, A(I)(d)−→− →A (I)(d) A(I)(d)/FI−→− →A(I)(d)/− → FI, where they are defined by ↦−→ +....

CalculateFd(ZK, m)/Fd+1(ZK, m), letting m be a local move such as (1) a # move, (2) a pass move, (3) a ∆ move, (4) an n-gon move....

(Y. Ohyama) Find necessary and sufficient conditions for two µ -component links ( µ > 2) to be ∆ link homotopic....

Let R be a commutative ring with 1, say, Z or Q. (1) Describe the spaces Fl(R(M K); loop)/Fl+1(R(M K); loop). (2) Describe the spaces Fl(RK; ∆ ∆) /Fl+...

(A. Kricker) Take (M1, K1) and (M2, K2) of the above sort. Then, there exists a (Z/pZ)-equivariant isomorphism φ: H1(Σ p (M1,K1); Z) → H1(Σ p (M2,K2);...

The map (15) is an isomorphism. This conjecture might be reduced to Conjecture 2.2 and the fo llowing conjec- ture....

{K∼ Cd O}/∼ Cd+1 is torsion free for each d....

(K. Habiro) Describe the abelian group {(M, K)∼ H Ld (S3, unknot)}/ ∼ H Ld+1 for each d....

(D. Bar-Natan) Is there a similar statement for finite type invariants of links? Let I be an ideal in the algebra V of finite type invariants of links...

(M.-J. Jeong, C.-Y. Park) Find a minimal finite subset An of Vn such that span (An) = Vn....

For each oriented knot K, calculate the Kontsevich invariant Z(K) for all degrees....

The Kontsevich invariant distinguishes oriented knots. (S ee Conjecture 2.5 for an equivalent statement of this conjectu re.)...

Does there exists a non-trivial oriented knot K such that Z(K) = Z(O) for the trivial knot O? (See Problem 2.6 for an equivalent problem.)...

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...

(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...