My question is about the paper "Affine tangles and irreducible exotic sheaves" (arxiv.org/abs/0802.1070). 


**Background:** Let $\mathcal{B}, \mathcal{N}$ denote the flag variety and nilpotent cone of $G=SL\_{2n}(\mathbb{C})$; let $T^\* \mathcal{B} = \tilde{\mathcal{N}}$ with projection map $p:T^\*\mathcal{B} \rightarrow \mathcal{B}$. Let $P\_k$ be the space of partial flags with the $k$-th flag omitted; let $V\_k$ be the tautological vector bundle on $\mathcal{B}$ of dimension $k$, and let $\mathcal{E}\_k$ be the pullback under $p$ of the quotient $V\_k/V\_{k-1}$.


For $z \in \mathcal{N}$, let $B\_{z}$ denote the Springer fibre corresponding to $z$ (i.e. the preimage of $z$ under the Springer resolution map $\pi: \tilde{\mathcal{N}} \rightarrow \mathcal{N}$). Let $S\_z$ be the Slodowy slice for $z$, and let $U\_z = \pi^{-1}(S\_z \cap \mathcal{N})$. Let $D\_z = D^b\_{B\_z}(U\_z)$ be the derived category of coherent sheaves on $U\_z$ supported on $B\_z$. If $z = Z\_{2n}$ is the nilpotent matrix of size $2n$ with Jordan type $(n,n)$, define $D\_{2n} = D\_z$.


Also define $P\_{k,z}$ to be the subvariety of $P\_k$ consisting of all flags $$(V\_0 \subset V\_1 \subset \cdots \subset V\_{k-1} \subset V\_{k+1} \subset \cdots \subset V\_n)$$ satisfying $\text{dim} V\_i = i$, $zV\_{i} \subset V\_{i-1}$ for $i \neq k,k+1$, and 
$zV\_{k+1} \subset V\_{k-1}$. 


**Q1** Could someone suggest a reference for framed tangles? Specifically, at the start of Section 3.3, I don't understand what "blackboard framing" means, or what "positive/negative twists of the framing of the i-th strand", means. 


**Q2** My question is how the functor $G\_{2n}^k: D\_{2m} \rightarrow D\_{2n}$ is defined (cf. bottom pg $10$, (1)). It is originally defined on pages $3-4$, as a functor $D^b(T^\*P\_k) \rightarrow D^b(\tilde{\mathcal{N}})$. Specifically, if $\pi\_k: T^\*P\_k \times\_{P\_k} \mathcal{B} \rightarrow T^\*P\_k$ is the projection, and $$i\_k: T^\*P\_k \times\_{P\_k} \mathcal{B} \rightarrow \tilde{\mathcal{N}}$$ be the embedding; then $G\_{2n}^k = \mathcal{E}\_k \otimes (i\_k)\_\* p\_k^\*$. But the domain and image of this map are different, so I'm confused. I thought perhaps Lemma $5$ on pg $10$, that $P\_{k, Z\_{2n}} \simeq B\_{Z\_{2n-2}}$ may be useful, but I'm still not quite sure.



Springer fibres for nilpotents of type $(n,n)$; framed tangles
For Q1: you should think of a framed tangle as a tangle where the strands are ribbons (you'll also see them called ribbon tangles). The blackboard framing of a ribbon tangle where the ribbon is pressed flat against the paper with no twist (so that twists can be indicated by doing a Reidemeister I move). Positive and negative twists means taking the ribbon and physically twisting it (positive means the direction that a human's right hand can comfortably twist).


For Q2: I believe she's restricting the functor to $U\_z$ and using the isomorphism you mention, but Jim is right that you might just want to ask Rina.


