Mathematics Problem Archive

(J. Roberts) Study the relation between the Jones polynomial and Gromov-Witten theory....

(X.-S. Lin) Describe the set of zeros of the Jones polynomial of all (alternating) knots. -1 -0.5 0.5 1 1.5 -1 -0.5 0.5 1 -1 -0.5 0.5 1 -1 -0.5 0.5 1 ...

(N. Dunfield) Find the relationship between the hyperbolic volume of knot complements and log VK (−1) (resp. log VK(−1)/ log degVK(t)). 3.5 4 4.5 5 5....

Understand Khovanov’s categorification of the Jones polyno - mial....

Categorify other knot polynomials....

(A. Stoimenow) Does the Jones polynomial V admit only finitely many values of given span? What about the Q polynomia l or the skein, Kauffman polynomia...

(A. Stoimenow) Why are the unit norm complex numbers α for which the value QK (α) has maximal norm statistically concentrated around e11π√ −1/25?...

(M. Kidwell, A. Stoimenow) Let K be a non-trivial knot, and let WK be a Whitehead double of K. Is then degm PWK (l, m) = 2 deg z FK (a, z) + 2?...

(E. Ferrand, A. Stoimenow) Is for any alternating link L, σ(L)≥ min degl ( PL(l, m) ) ≥ min dega ( FL(a−1, z) )?...

(A. Stoimenow) If∇k is the coefficient of zk in the Conway polynomial and c(L) is the crossing number of a link L, is then ⏐ ⏐∇k(L) ⏐ ⏐≤ c(L)k 2k k!?...

(A. Stoimenow) Does min deg a ( FL(a−1, z) ) ≤ 1− χ(L) hold for any link L? If u(K) is the unknotting number of a knot K, does min dega ( FK (a−1, z) ...

(The volume conjecture, [198, 296]) For any knot K, 2π·lim N →∞ log|JN (K)| N = v3||S3− K||, (2) where||·||denotes the simplicial volume and v3 denote...

Justify the above arguments rigorously....

(H. Murakami, J. Murakami, M. Okamoto, T. Takata, Y. Yokota [297]) For a hyperbolic link L, 2π √ −1·lim N →∞ log JN (L) N = CS(S3− L) + √ −1vol(S3− L)...

(H. Murakami) For a torus knot K, calculate CS(S3− K) (giving an appropriate definition of it) and calculate lim log JN (K) N (fixing an appropriate c...

([220, Problem 1.92 (N)]) Fd(ZK)/Fd+1(ZK) is torsion free for each d....

A(S1; Z) is torsion free....

(X.-S. Lin [262]) Let R be a commutative ring with 1, say Z/2Z. Every weight system A(S1; R)(d)/FI→ R is induced by some Vassiliev invariant RK→ R....

(T. Stanford) The Dogolazky-Kneissler 2-torsion element in A(↓↓, Z) (see Figure 7) can be embedded into a chord diagram in A(S1, Z) in many ways. Such...

Vassiliev invariants distinguish oriented knots. (See Con jec- ture 3.2 for an equivalent statement of this conjecture.)...

Does there exists a non-trivial oriented knot which can not b e distinguished from the trivial knot by Vassiliev invariant s? (See Problem 3.3 for an ...

(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 ⌋....

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....