Annular Khovanov homology and knotted Schur-Weyl representations

作者：Grig**y J Elisenda; Licata Anthony M; Wehrli Stephan M

来源：Compositio Mathematica, 2018, 154(3): 459-502.

DOI：10.1112/S0010437X17007540

<jats:p>Let <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0010437X17007540_inline1" xlink:type="simple" />
<jats:tex-math>$\mathbb{L}\subset A\times I$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0010437X17007540_inline2" xlink:type="simple" />
<jats:tex-math>$\mathfrak{sl}_{2}(\wedge )$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, the exterior current algebra of <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0010437X17007540_inline3" xlink:type="simple" />
<jats:tex-math>$\mathfrak{sl}_{2}$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. When <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0010437X17007540_inline4" xlink:type="simple" />
<jats:tex-math>$\mathbb{L}$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is an <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0010437X17007540_inline5" xlink:type="simple" />
<jats:tex-math>$m$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>-framed <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0010437X17007540_inline6" xlink:type="simple" />
<jats:tex-math>$n$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>-cable of a knot <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0010437X17007540_inline7" xlink:type="simple" />
<jats:tex-math>$K\subset S^{3}$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, its sutured annular Khovanov homology carries a commuting action of the symmetric group <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0010437X17007540_inline8" xlink:type="simple" />
<jats:tex-math>$\mathfrak{S}_{n}$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. One therefore obtains a ‘knotted’ Schur–Weyl representation that agrees with classical <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0010437X17007540_inline9" xlink:type="simple" />
<jats:tex-math>$\mathfrak{sl}_{2}$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> Schur–Weyl duality when <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0010437X17007540_inline10" xlink:type="simple" />
<jats:tex-math>$K$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is the Seifert-framed unknot.</jats:p>

• 出版日期2018-3