论文标题
Khovanov同源性和两次圆环的无可纹身角色品种的福卡亚类别
Khovanov homology and the Fukaya category of the traceless character variety for the twice-punctured torus
论文作者
论文摘要
我们描述了一种策略，用于通过概括降低Khovanov同源性的链接解释来构建镜头空间中的链接的策略，这是由于Hedden，Herald，Hogancamp和Kirk引起的$ S^3 $链接。该策略依赖于对福卡亚（Fukaya）类别的部分猜想描述，$ su（2）$两道刺的字符品种都有两次穿刺。从固体圆环中的1键键的图中，我们在此福卡亚类别的$ a_ \ infty $类别中构造了相应的对象$（x，δ）$。 $（x，δ）$的同型类型是缠结图的同位素不变。我们使用$（x，δ）$来构建$ s^3 $中链接的Cochain Complexse，以及$ S^2 \ times s^1 $中的某些链接。对于$ S^3 $中的链接，我们的Cochain Complex的共同体学再现了降低的Khovanov同源性，尽管Cochain Complex本身并不是通常的。对于$ s^2 \ times s^1 $中的链接，我们提出的结果表明，我们的Cochain Complex的共同体可能是链接不变的。
We describe a strategy for constructing reduced Khovanov homology for links in lens spaces by generalizing a symplectic interpretation of reduced Khovanov homology for links in $S^3$ due to Hedden, Herald, Hogancamp, and Kirk. The strategy relies on a partly conjectural description of the Fukaya category of the traceless $SU(2)$ character variety of the 2-torus with two punctures. From a diagram of a 1-tangle in a solid torus, we construct a corresponding object $(X,δ)$ in the $A_\infty$ category of twisted complexes over this Fukaya category. The homotopy type of $(X,δ)$ is an isotopy invariant of the tangle diagram. We use $(X,δ)$ to construct cochain complexes for links in $S^3$ and some links in $S^2 \times S^1$. For links in $S^3$, the cohomology of our cochain complex reproduces reduced Khovanov homology, though the cochain complex itself is not the usual one. For links in $S^2 \times S^1$, we present results that suggest the cohomology of our cochain complex may be a link invariant.
