17 problems

Let $D$ be a diagram of an oriented framed link $L$ whose components or strands are colored by numbers from $1$ to $n-1$, and let $P_n(L)$ be the corresponding colored link invaria…

Let $D_{n,m}^i$ and $E_{n,m}^i$ be the diagrams defined in the paper, and consider the saddle cobordism … at the crossing corresponding to the last letter $\sigma_i$. Surjectivity…

Equality conjecture. If each component of $F$ intersects $M_1$, then, as $\mathbb{Z}$-graded bimodules over $(A_{EF}(M_2),A_{EF}(M_1))$,

Let $w(D)$ denote the word invariant associated with a diagram $D$ of a knot with two parities. A sliceness obstruction conjecture. $w(D)$ is a sliceness obstruction for $2$-knots…

Let $B\mathrm{Cob}_2 \simeq S^1 \times X$ be Tillmann's equivalence of infinite loop spaces, where $X$ is a simply connected infinite loop space. A classifying space is a 1-type if…

Let $K\subset S^3$ be a knot, let $S^3_{-n}(K)$ be obtained by $(-n)$-surgery, and let $W$ be the natural cobordism from $S^3$ to $S^3_{-n}(K)$. Let $t_i:H_2(W_{-n})\to\mathbb{Z}$…

Let ${\cal C\mkern-1.1mu o\mkern-0.2mu b}^{2+1}$ be the category of relative 3-dimensional cobordisms, and let $\overline{\cal Alg}$ be the algebraic category generated by a Hopf a…

Let $K$ be a knot in $S^3$. A concordance is a cobordism of minimal intrinsic complexity, and it has minimal extrinsic complexity when it has no index $2$ Morse critical points. Sl…

Let $(M,\gamma)$ be a balanced sutured manifold, and let $\mathcal{W}=(W=M\times[0,1],Z=\partial M\times[0,1],[\xi])$ be the trace sutured cobordism from…

Let $(M_1,gamma_1)$ and $(M_2,gamma_2)$ be balanced sutured manifolds, and let $(W,Z,[\xi])$ be sutured cobordism data, where $Z$ is equipped with a contact structure whose boundar…

Let $M$ be a hyperbolic $4$-cobordism with boundary components $N_0$ and $N_1$. Suppose that $N_0$ and $N_1$ are highly symmetric to each other, both topologically and geometricall…

Grid-cobordism functoriality conjecture. Grid diagram maps induce maps on $\operatorname{HFK}^-$ that are invariant with respect to marked smooth isotopy classes of surfaces.

The Khovanov homotopy type assigns a spectrum to a link, and cobordism maps on Khovanov spectra are defined by composing maps associated to elementary cobordisms. Decomposition-ind…

Let $M$ be a multigraph, and let $\mathbf{TCircD}_M$ be the free braided strict monoidal category on $M$ in which each object of $M$ is equipped with a symmetric Frobenius algebra…

A unitary extended TQFT is a map between stable $n$-categories. Let $n\operatorname{Cob}_n$ denote the stable $n$-category of framed cobordisms, with objects compact framed $0$-dim…

Let $(A,E)$ be a commutative Frobenius pair over $k$. Let ${\bf FP}(A,E)$ be the subcategory of ${\rm Mod}_k$ whose objects are generated by $A^{\otimes m}\otimes E^{\otimes n}$ fo…

Classifying-space conjecture. We conjecture that the constructed space is the classifying space of the category of embedded 5-cobordisms.