In this paper we construct invariants of 3-manifolds "à la Reshetikhin-Turaev" in the setting of non-semi-simple ribbon tensor categories. We give concrete examples of such categories which lead to a family of 3-manifold invariants indexed by the integers. We prove this family of invariants has several notable features, including: they can be computed via a set of axioms, they distinguish homotopically equivalent manifolds that the standard Reshetikhin-Turaev-Witten invariants do not, and they allow the statement of a version of the Volume Conjecture and a proof of this conjecture for an infinite class of links.
We study complexities of 3-manifolds defined from triangulations, Heegaard splittings, and surgery presentations. We show that these complexities are related by linear inequalities, by presenting explicit geometric constructions. We also show that our linear inequalities are asymptotically optimal. Our results are used in [arXiv:1405.1805] to estimate Cheeger-Gromov $L^2$ $ρ$-invariants in terms of geometric group theoretic and knot theoretic data.
Reshetikhin-Turaev invariants of Seifert 3-manifolds and a rational surgery formula
我们计算RT不变量的所有定向塞弗特流形直接从手术介绍。我们在任意模范畴的一般框架中工作，如[V.G. Turaev，Quantum invariants of knots and 3--manifold，de Gruyter Stud.Math.18，Walter de Gruyter（1994）]，这些不变量用模范畴的S-矩阵和T-矩阵表示。在另一个方向，我们推导出一个合理的手术公式，它说明了RT-不变量的行为下，合理的手术沿框架链接在任意封闭的定向3-流形嵌入彩色丝带图。利用外科手术公式给出了具有可定向基的Seifert流形的RT-不变量的另一种推导方法。
This paper describes a Dehn surgery approach to generating asymmetric hyperbolic manifolds with two distinct lens space fillings. Such manifolds were first identified in work of Dunfield-Hoffman-Licata as the result of a computer search of the SnapPy census, but the current work establishes a topological framework for constructing vastly many more such examples. We introduce the notion of a ``jointly primitive'' presentation of a knot and show that a refined version of this condition ``longitudinally jointly primitive'' is equivalent to being surgery dual to a $(1,2)$--knot in a lens space. Th…
Combinatorial Dehn-Lickorish Twists and Framed Link Presentations of 3-Manifolds Revisited
由任意闭可定向连通三维流形的$n$四面体$T$的伪三角剖分（简称{ \em a 3D-space}）$M^3$，我们提出了一个gem $J '$，它诱导出$\IS^3$，具有以下特征：（a）它的顶点数为O（n）：（b）它有一组$p$两两不相交的顶点对$\{u_i,v_i\}$，每个顶点对称为{ \em a twistor};（c）在$J '$的对偶$(J ')^\star$中，扭量变成一对四面体，其具有一对公共的相对边，并且它被命名为{ \em a铰链};（d）在$(J ')^\star \subset \IS^3$的任何嵌入中，每个铰链的$ε$ -邻域是实心环面;（e）这些$p$实心环面是成对不相交的;（f）每个扭曲器包含关于如何基于对应于它的实心环面上的Denh-Lickorish扭曲来执行特定手术的精确描述;（g）进行所有这些$p$手术（在双宝石的水平上）我们产生具有$|G '|=M^3$的宝石$G '$;（h）在$G '$中，每个这样的手术通过交换每对顶点中的一对邻居来完成：特别是$|V(G ')=|V(J ')|$。这是Wallace（1960）和Lickorish（1962）的经典定理的新证明，{ \em based on a linear polynomial algorithm}，即每个3D空间在$\IS^3$中具有框架链接表示，并为通过$O(n^2)$算法实际获得…
We prove that the stable moduli space of $(n-1)$-connected, $n$-parallelizable, $(2n+1)$-dimensional manifolds is homology equivalent to an infinite loopspace for $n \geq 4, n \neq 7$. The main novel ingredient is a version of the cobordism category incorporating surgery data in the form of Lagrangian subspaces.
We give a diagrammatic presentation of the category of $\textbf{U}_q(\mathfrak{sl}_2)$-tilting modules $\mathfrak{T}$ for $q$ being a root of unity and introduce a grading on $\mathfrak{T}$. This grading is a "root of unity phenomenon" and might lead to new insights about link and $3$-manifold invariants deduced from $\mathfrak{T}$. We also give a diagrammatic category for the (graded) projective endofunctors on $\mathfrak{T}$, indicate how our results could generalize and collect some "well-known" facts to give a reasonably self-contained exposition.
