Mathematics Problem Archive

(M. Polyak, see [153, “Theorem 4”]) Let F be an oriented compact surface. Two homology cylinders C and C ′ over F are Yd -equivalent if and only if v(...

(F. Deloup) Classify the monoid (for orthogonal sum) of isomorphism classes of quadratic forms qσ....

(G. Massuyeau) Describe the quotient set {spin closed 3-manifolds}/∼ Y s d, in particular, for d = 2, 3....

(F. Deloup, G. Massuyeau) Describe the quotient set {spin c closed 3-manifolds}/∼ Y c d, in particular, for d = 2, 3....

(F. Deloup) Is there a lift of arg γ(qσ) to a mod 16 invariant? This would give a finite type invariant of degree 1 in the spin c Goussarov-Habiro the...

For each rational homology 3-sphere M, calculate Z L M O(M ) for all degrees....

The LMO invariant distinguishes integral homology 3-spheres. (See Conjecture 10.9.)...

Does there exist an integral/rational homology 3-sphere M such that Z L M O(M ) = Z L M O(S3)? 11.3 Characterization of the image of the LMO invariant...

Characterize those elements of ˆA(∅)conn which are of the form log Z L M O(M ) for integral/rational homology 3-spheres....

Construct the LMO invariant with coefficients in a finite field....

Construct the LMO invariant (or the theory of finite type invariants) in arrow diagrams. 11.5 Refinements of the LMO invariant (T. Le) As mentioned in...

(T. Le, V. Turaev) Define the LMO invariant Z L M O(M, σ) of the pair of a closed 3-manifold M and a spin structure σ of M such that Z L M O(M ) = ∑ σ...

(T. Le, V. Turaev) For every element ξ∈ H 1(M, Z) construct an extension of Z L M O(M, ξ) of the LMO invariant such that when ξ = 0 one recovers the u...

(1) Find a surgery formula for the Kuperberg-Thurston in- variant [237] in terms of the Chern-Simons series of Questio n 3.12 (2) Compare the Kuperber...

(D. Thurston) Do configuration spaces of [237] have torsion in Z-homology? Does such torsion deduce a torsion invariant of h omology 3-spheres?...

(R. Benedetti) Are torsions actually sensitive only to the (pL)-homotopy immersion classes of (pL)-knots? If one fix a C - homotopy immersion class of...

(R. Benedetti) For every W, for every (pL)-class α1 as above, f ∗ 1 is an isomorphism. This means, in particular, that finite typ e invari- ants of Le...

(H.R. Morton) From a knot diagram find an explicit such homomorphism to some permutation group or establish that th e knot is trivial. Refinements. (1...

(Y. Nakanishi, T. Harikae [220, Conjecture 1.59 (6)]) Any link can be related to a trivial link by a sequence of (2,2)-mo ves....

Find a new proof of the existence of a universal Vassiliev in- variant of knots, presenting them by KTG’s and their operati ons....

(D. Bar-Natan, D. Thurston) For each compact Lie group G, level k, and every KTG K: Γ → R3, there exists a collection of measures µ K on the space of ...

Construct an invariant of KTG’s from configuration space in- tegrals in a natural way. Turaev [388] introduced a presentation of 3-manifolds as S1 -bu...

(D. Thurston) The shadow number of a 3-manifold is quasi-linear in its Gromov norm. That is, there exist consta nts c1 and c2 such that c1||M||≤ (shad...

(D. Thurston) Find a condition on shadow diagrams which is satisfied by shadow diagrams from alternating knots; and g ives a lower bound on the hyperb...

Construct a universal Reshetikhin-Turaev invariant and a universal Turaev-Viro-Ocneanu invariant of closed 3-mani folds, in terms of the KTG algebra....

(J. Roberts) What are quantum groups?...

(N. Askitas) Can a knot of 4-genus gs always be sliced (made into a slice knot) by gs crossing switches?...

(M. Boileau [220, Problem 1.69 (C)]) Are there mutants of distinct unknotting numbers?...

(X.-S. Lin [262]) Any automorphism of G is either the identity or the mirror map, that is, any automorphism of G is induced by a diffeomorphism of the ...

(X.-S. Lin [262]) What is the homotopy type of the space L(K) of long ropes (as shown in the picture below) with the fixed kno t type K?...

(J. Roberts) Extend Kuperberg’s work on webs....

(J. Roberts) Extend the theory of measured laminations to higher rank groups....

(J. Roberts) What is the generating function for q -spin net evaluations?...

(Y. Shinohara [364]) If n = 4 k + 1 with k > 0, is there a knot with determinant n and signature 4?...

(T. Stanford) IsC2 solvable? Does C2 contain a free group?...

(A. Stoimenow) Do positive links of given signature σ have bounded (below) maximal Euler characteristic χ?...

(A. Stoimenow) If a prime knot K can be transformed into its mirror image by one crossing change, is K achiral or (algebraically?) slice?...

(A. Stoimenow) Let n be an odd natural number, different from 1, 9, and 49, such that n is the sum of two squares. Is there a prime alternating achiral...

(V. Turaev) A pair (a finitely generated abelian group H of rank 1, an element ∆( t)∈ Z[H/TorsH] = Z[t±1]) (where t is a generator of H/TorsH ) can be...

The Flat Hierarchy: The flat hierarchy is constructed for any ordinal $\alpha$. We label flat crossings with members of this ordinal. In a flat third ...

Virtual Three Manifolds: There is a theory of virtual $3$–manifolds constructed as formal equivalence classes of virtual diagrams modulo generalized K...

Virtual Biquandle: Construct presentations of the virtual biquandle with the a linear (non-commutative) representation at classical crossings and some...

The Fundamental Biquandle: Does the fundamental biquandle, see classify virtual links up to mirror images? (We know that the biquandle has the same va...

Virtualization and Unit Jones Polynomial: Suppose the knot $K$ is classical and not trivial. Suppose that ${\tilde K}$ (obtained from $K$ by virtualiz...

Virtual Quandle Homology: Study virtual quandle homology in analogy to quandle homology ....

Wild Virtuals: Create the category of “wild virtual knots” and establish its axiomatics. In particular, one needs a theorem that states when a wild eq...

Embeddings of Surfaces: Given a non-trivial virtual knot $K$. Prove that there exists a minimal realization of $K$ in $N=S_{g}\times I$ and an unknott...

Non-Commutativity and Long Knots: It is known that any classical long knot commutes with any long knot. This is definitely not the case in the virtual...

Find new geometric/topological interpretations for the Jones polynomial and for Khovanov homology....

Let $VKT/Z$ denote virtual knot theory modulo $Z$-equivalence, as defined in the section above on virtual knot theory. Recall that two virtual diagram...