Mathematics Problem Archive

Determine the dimension of the space of primitive Vassiliev invariants of each degree d. Equivalently, determine the dimension of the space A(S1; Q)(d...

(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(...

(M. Polyak) Milnor’s µ -invariants of string links can be de- fined similarly as above (see [329]). Find a topological pres entation of a µ - invarian...

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 ↦−→ +....

[154] Every Vassiliev invariant of classical knots can be extended to a finite type invariant of long virtual knots. (Se e also Problem 3.9.) 2.8 Fini...

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 [165], see also [153, “Theorem 5”]) Two m- strand string links L and L′ are Cd -equivalent if and only if v(L) = v(L′) for any A-valued fin...

(M. Polyak) Establish the Goussarov-Habiro theory for vir- tual knots....

(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.)...

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)....