Kirby elements and quantum invariants

标准曲面的远距和的 Kirby–Thompson 不变量

Kirby-Thompson invariants of distant sums of standard surfaces

本文计算标准曲面任意有限远距和的 L 与 L* 不变量，并给出桥数及两类不变量均可任意增大的纽结曲面首例。

Kirby calculus for null-homologous framed links in 3-manifolds

A theorem of Kirby gives a necessary and sufficient condition for two framed links in S^3 to yield orientation-preserving diffeomorphic results of surgery. Kirby's theorem is an important method for constructing invariants of 3-manifolds. In this paper, we prove a variant of Kirby's theorem for null-homologous framed links in a 3-manifold. This result involves a new kind of moves, called IHX-moves, which are closely related to the IHX relation in the theory of finite type invariants. When the first homology group of M is free abelian, we give a refinement of this result to \pm1-framed, algebra…

Evaluating TQFT invariants from G-crossed braided spherical fusion categories via Kirby diagrams with 3-handles

A family of TQFTs parametrised by G-crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the resulting manifold invariants are scarce because of the combinatorial complexity of triangulations, if nothing else. Handle decompositions, and in particular Kirby diagrams are known to offer an economic and intuitive description of 4-manifolds. We show that if 3-handles are added to the picture, the state sum model can be conveniently redefined by translating Kirby diagrams into the graphical calculus of…

Hopf diagrams and quantum invariants

The Reshetikhin-Turaev invariant, Turaev's TQFT, and many related constructions rely on the encoding of certain tangles (n-string links, or ribbon n-handles) as n-forms on the coend of a ribbon category. We introduce the monoidal category of Hopf diagrams, and describe a universal encoding of ribbon string links as Hopf diagrams. This universal encoding is an injective monoidal functor and admits a straightforward monoidal retraction. Any Hopf diagram with n legs yields a n-form on the coend of a ribbon category in a completely explicit way. Thus computing a quantum invariant of a 3-manifold r…

4维2柄体中ribbon表面的量子不变量

Quantum Invariants of Ribbon Surfaces in $4$-Dimensional $2$-Handlebodies

本文通过单模ribbon范畴构建4维2柄体中ribbon表面的量子不变量，并恢复多个已有不变量，同时推广Reshetikhin-Turaevfunctor到标记Kirby图类，产生带框架链接的不变量。

从三重分解图定义4维流形不变量

Categorical 4-manifold invariants from trisection diagrams

本文通过三重分解图定义新的4维流形不变量，结合双模范畴和pivotal函数，恢复了霍普夫代数和模融合范畴的不变量。

Cyclotomic expansions for $\mathfrak{gl}_N$ knot invariants via interpolation Macdonald polynomials

In this paper we construct a new basis for the cyclotomic completion of the center of the quantum $\mathfrak{gl}_N$ in terms of the interpolation Macdonald polynomials. Then we use a result of Okounkov to provide a dual basis with respect to the quantum Killing form (or Hopf pairing). The main applications are: 1) cyclotomic expansions for the $\mathfrak{gl}_N$ Reshetikhin--Turaev link invariants and the universal $\mathfrak{gl}_N$ knot invariant; 2) an explicit construction of the unified $\mathfrak{gl}_N$ invariants for integral homology 3-spheres using universal Kirby colors. These results …

Weak Hopf Algebras unify the Hennings-Kauffman-Radford and the Reshetikhin-Turaev invariant

We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. This quantum trace satisfies conditions that render the link evaluation invariant under Kirby moves. If H is a suitable finite-dimensional Hopf algebra (not weak), our invariant reduces to the Kauffman-Radford invariant for the dual of H. If H is the Weak Hopf Algebra Tannaka-Krein …

接触切割图与Weinstein $\mathcal{L}$-不变量

The contact cut graph and a Weinstein $\mathcal{L}$-invariant

本文定义接触切割图并建立其与Lefschetz纤维化及带divides多截面的对应，由此构造Weinstein域的辛不变量$\mathcal{L}$，讨论其与Lefschetz稳定化的关系，并给出$\mathcal{L}=0$的约束条件及$\mathcal{L}$任意大的例子。

Finite Type Enhancements

We enhance the biquandle counting invariant using elements of truncated biquandle-labeled Polyak algebras. These finite type enhancements reduce to the finite type enhancements defined by Goussarov, Polyak and Viro for the trivial biquandle of one element and determine (but are not determined by) the biquandle counting invariant for general biquandles. Unlike the unlabeled case, biquandle labeled finite type invariants of degree 1 are nontrivial and are related to biquandle cocycle invariants.

On the non-existence of elements of Kervaire invariant one

We show that Kervaire invariant one elements in the homotopy groups of spheres exist only in dimensions at most 126. By Browder's Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. With the exception of dimension 126 this resolves a longstanding problem in algebraic topology.

On transverse invariants from Khovanov homology

O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two refinements of Plamenevskaya's invariant, one valued in Bar-Natan's deformation of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum. We show that the first of these refinements is invariant under negative flypes and SZ moves; this implies that Plamenevskaya's class is also invariant under these moves. We go on to show that for small-crossing transverse knots K, both refinements are determined by the classical invariants of K.

关于纽结格同调中的勒让德不变量

On the Legendrian invariant in knot lattice homology

研究纽结格同调中的勒让德不变量，通过将正规曲面奇点链环的接触不变量转换到格同调理论，利用亚历山大分次定义元素\(\mathcal{L}(L)\)，证明其在爆破下不变，并借助链复形的滤过链同伦给出勒让德不变量的部分组合描述。

Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups

Let $F_n$ be the free group on $n$ generators and $Γ_g$ the surface group of genus $g$. We consider two particular generating sets: the set of all primitive elements in $F_n$ and the set of all simple loops in $Γ_g$. We give a complete characterization of distorted and undistorted elements in the corresponding $Aut$-invariant word metrics. In particular, we reprove Stallings theorem and answer a question of Danny Calegari about the growth of simple loops. In addition, we construct infinitely many quasimorphisms on $F_2$ that are $Aut(F_2)$-invariant. This answers an open problem posed by Mikló…