18 problems

Let $\Sigma$ be a connected oriented surface, let $n,k\geq 0$, and consider $n$-string links in $\Sigma\times[0,1]$. Two such string links are $C_{k+1}$-equivalent when they are re…

Let $L(m)$ denote the string links on $m$ strands. For $n\geq 1$, let $V_n$-equivalence be the equivalence relation induced by the Vassiliev filtration, and let $C_n$-equivalence b…

The $w_k$-equivalence conjecture. Two welded string links are $w_k$-equivalent if and only if they cannot be distinguished by finite type invariants of degree $<k$.

Fix $m\geq 2$, $k\geq 1$, and $n\geq 3$. Let … be the graphing map. Graphing injectivity conjecture. The induced maps on rational homotopy and rational homology are injective: … ar…

Fix $m\geq 2$, $k\geq 1$, and $n\geq 3$. Let $\operatorname{Conf}(m,\mathbb R^n)$ be the configuration space of $m$ points in $\mathbb R^n$, and let … be the graphing map. Real gra…

Let $m\geq 2$ and $n\geq 3$, and consider the rational homotopy groups of the configuration space $\operatorname{Conf}(m,\mathbb R^n)$. Graphing extension conjecture. The injectivi…

Let $m\geq 2$, $n\geq 3$, and let $G$ denote the graphing map from the based loop space of the configuration space of $m$ points in $\mathbb R^n$ to the corresponding space of high…

Consider the sum of the trees depicted in Figure 8 of the source, associated to 2-component string links, with coefficients in $\mathbb Z/2\mathbb Z$. Framed Arf finite-type conjec…

Lower-range hairy graph complex conjecture. The same conclusion should hold whenever

Let $k>2$ and consider the spaces and Swiss Cheese operad action for $k$-stranded string links analogous to those for string $2$-links. The relevant adaptations of Theorems and are…

Two links are $C_k$-equivalent when they are connected by a finite sequence of simple tree clasper surgeries of degree at least $k$, where the degree of a clasper is half the numbe…

Let $O_{1},\ldots,O_{k}$ be well-pointed, $\Sigma$-cofibrant, weakly $2$-reduced operads relative to another weakly $2$-reduced operad $O$. Let $\eta:\mathbb{O}^{+}\rightarrow M$ b…

Goodwillie--Weiss truncation homotopy conjecture. If $d-\max\{m_i\mid 1\leq i\leq r\}>2$, then

Low-codimension rational homotopy splitting conjecture. For every degree $*\geq 0$ and every $d-\max\{m_i\mid 1\leq i\leq r\}>2$, the identity giving the rational homotopy splittin…

Homotopy graph-complex conjecture. The homotopy graph-complex computes the rational homotopy groups of the space of links whenever the codimension is greater than two, namely whene…

Let $\mathrm{pStr}(n)$ be the monoid of pure string links. The Kirk–Livingston–Wang representation is … and let $p\mathcal W_{n,1/0}$ denote the corresponding quotient of the two-v…

The authors define Conway-type finite-type invariants for ordered tangles, extending the coefficients of the Conway polynomial to virtual tangles. For string links, a shifted order…

Let $FT_n$ denote the equivalence relation on tangles determined by finite-type equivalence up to degree $n$, so that $FT_n$-equivalent tangles differ by an element of the $(n+1)$s…