Perverse Sheaves and Knot Contact Homology
Perverse Sheaves and Knot Contact Homology
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反常滑轮和结接触同源性
DOI：
10.1016/j.crma.2017.02.007
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Wai
中科院分区：
文献类型：
--
作者：
Y. Berest;A. Eshmatov;Wai
在一系列论文[35-39]中，L. Ng引入并研究了由半自由微分梯度（DG）代数AL表示的R3中链路L的一个新的代数不变量。该DG代数的结构（称为组合结DGA）由表示链路L的编织群Bn中的一个元素决定。AL的同调称为结接触同调HC * (L)。因为它伴随着Legendrian接触相同的第1单元余法线包L⊆圣∗R3 L .这个巧合的推测(35、36)和证明后(13、14),在那里,事实上,整个组合结DGA同构DG定义的几何代数计算L.Our Legendrian接触相同的原始动机是理解Ng的组合的不变性证明艾尔(quasi-isomorphism)马尔可夫下动作。值得注意的是，虽然在[35]中用显式公式定义了AL的微分，但其组合结构相当复杂，其代数起源似乎很神秘。即使AL的第0次同调是一个链路不变量，从[35]的定义来看也远不明显(参见。[35，第4.3节])。因此，我们提出了一个不同的，更概念化的结构，使AL的马尔可夫不变性相当透明2，更重要的是，将结接触同调与其他经典不变量，如结群和亚历山大模块放在一行。
In a series of papers [35–39], L. Ng introduced and studied a new algebraic invariant of a link L in R3 represented by a semi-free differential graded (DG) algebra AL. The structure of this DG algebra (termed a combinatorial knot DGA) is determined by an element of a braid group Bn representing the link L. The homology of AL is called the knot contact homology HC∗(L), as it coincides with the Legendrian contact homology 1 of the unit conormal bundle L⊆ ST∗ R3 of L. This coincidence was conjectured in [35, 36] and proved later in [13, 14], where it was shown, in fact, that the entire combinatorial knot DGA is isomorphic to a geometrically defined DG algebra computing the Legendrian contact homology of L.Our original motivation was to understand Ng’s combinatorial proof of the invariance of AL (up to quasi-isomorphism) under the Markov moves. We should remark that, although the differential of AL is defined in [35] by an explicit formula, its combinatorial structure is fairly complicated and its algebraic origin seems mysterious. Even the fact that the 0-th homology of AL is a link invariant is far from being obvious from the definition of [35](cf.[35, Section 4.3]). As a result, we have come up with a different, more conceptual construction that makes the Markov invariance of AL quite transparent 2 and, more importantly, places knot contact homology in one row with other classical invariants, such as knot groups and Alexander modules.
影响因子：
2.6
作者：
Anno, Rina;Logvinenko, Timothy
通讯作者：
Logvinenko, Timothy