Mathematics Problem Archive

(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...

(3-move conjecture, Y. Nakanishi [305]) Any link can be related to a trivial link by a sequence of 3-moves....

(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...

Recognising the Kishino Knot: There have been invented many ways to recognize the Kishino virtual knot (from the unknot): The $3$–strand Jones polynom...

Flat Virtuals: Flat virtual knots, also known as virtual strings , are difficult to classify. Find new combinatorial invariants of flat virtual knots....

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 ...

Virtuals and the Theory of Doodles: Compare flat theories of virtual knots with theories of doodles. A doodle is represented by a flat diagram in the ...

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

Welded Knots: We would like to understand welded knots . It is well known that if we admit forbidden moves to the virtual link diagrams, each virtual ...

Long Knots and Long Flat Knots: Enlarge the long knot invariant structure proposed in . Can one get new classical knot invariants from the approach in...

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

Virtual braids: Is there a birack such that its action on virtual braids is faithful? Is the invariant of virtual braids in , see also , faithful? T...

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 ....

Khovanov Homology: Construct a generalization of the Khovanov complex for the case of virtual knots that will work for arbitrary virtual diagrams. Inv...

Brauer algebra: The appropriate domain for the virtual recoupling theory is to place the Jones–Wenzl projectors in the Brauer algebra. That is, when w...

Virtual Alternating Knots: Define and classify alternating virtual knots. Find an analogue of the Tait flyping conjecture and prove it. Compare . Cl...

Crossing Number: One of the most important problems in knot theory is the problem of finding the minimal crossing number for a given knot. It can be e...

Crossing number problems: For each virtual link $L$, there are three crossing numbers: the minimal number $C$ of classical crossings, the minimal numb...

Vassiliev Invariants: Understand the connection between virtual knot polynomials and the Vassiliev knot invariants of virtual knots (in Kauffman's sen...

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...

The Rack Space: The rack space was invented by Fenn, Rourke and Sanderson . The homology of the rack space has been considered by the above authors an...

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