Aleksandar Mijatovic
It is not completely unreasonable to expect that a computable function bounding the number of Pachner moves needed to change any triangulation of a given 3-manifold into any other triangulation of the same 3-manifold exists. In this paper we describe a procedure yielding an explicit formula for such a function if the 3-manifold in question is a Seifert fibred space.
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We prove that there are exactly $6$ Nil Seifert fibred spaces which can be obtained by Dehn surgeries on non-trefoil knots in $S^3$, with $\{60, 144, 156, 288, 300\}$ as the exact set of all such surgery slopes up to taking the mirror images of the knots. We conjecture that there are exactly $4$ specific hyperbolic knots in $S^3$ which admit Nil Seifert fibred surgery. We also give some more general results and a more general conjecture concerning Seifert fibred surgeries on hyperbolic knots in $S^3$.
We show that if $B$ is an aspherical 2-orbifold in one of the families known to have orbifold fundamental groups of weight 1 then $B$ is the base of a Seifert fibration of a 2-knot manifold $M(K)$.
For every genus $g\geq 2$, we construct an infinite family of strongly quasipositive fibred knots having the same Seifert form as the torus knot $T(2,2g+1)$. In particular, their signatures and four-genera are maximal and their homological monodromies (hence their Alexander module structures) agree. On the other hand, the geometric stretching factors are pairwise distinct and the knots are pairwise not ribbon concordant.
We consider the question of when is the closed manifold obtained by elementary surgery on an $n$-knot Seifert fibred over a 2-orbifold. After some observations on the classical case, we concentrate on the cases n=2 and 3. We have found a new family of 2-knots with torsion-free, solvable group, overlooked in earlier work. We know of no higher dimensional examples.
The present paper gives an example of a rigid spherical cone-manifold and that of a flexible one which are both Seifert fibred.
Alexander polynomial, Dijkgraaf-Witten invariant, and Seifert fibred surgery
本文利用 Dijkgraaf-Witten 不变量研究沿纽结的 k/ℓ 手术产生的小 Seifert 3-流形，证明其重数与手术系数受 Alexander 多项式约束。
We introduce a new algebraic topological technique to detect non-fibred knots in the three sphere using the twisted Alexander invariants. As an application, we show that for any Seifert matrix of a knot with a nontrivial Alexander polynomial, there exist infinitely many non-fibered knots with the given Seifert matrix. We illustrate examples of knots that have trivial Alexander polynomials but do not have twisted Alexander invariants of fibred knots.
We consider Chern-Simons theory with complex gauge group and present a complete non-perturbative evaluation of the path integral (the partition function and certain expectation values of Wilson loops) on Seifert fibred 3-Manifolds. We use the method of Abelianisation. In certain cases the path integral can be seen to factorize neatly into holomorphic and anti-holomorphic parts. We obtain closed formulae of this factorization for the expectation values of torus knots.
Fillable structures on negative-definite Seifert fibred spaces
本文研究负定星形连接空间上的可填充接触结构，证明此类Seifert纤维空间具有唯一的负最大扭数，并通过晶格上同调的Alexander滤子计算该扭数，进一步展示这些流形上的负扭紧结构由复表面奇点的最小解析中的Stein结构诱导。
We prove that basket links, whose symmetrized Seifert form is congruent to the Cartan matrix of the simply laced Dynkin diagram $A_{n}$, are isotopic to the torus link $T(2,n+1)$. In addition, we provide examples of links, constructed by plumbing $n$ positive Hopf bands the core curves of which intersect at most once, with symmetrized Seifert form congruent to the Cartan matrix $A_{n}$, that are not isotopic to $T(2,n+1)$.
One measure of the complexity of a 3-manifold is its triangulation complexity: the minimal number of tetrahedra in a triangulation of it. A natural question is whether we can relate this quantity to its topology. We determine the triangulation complexity of Seifert fibered spaces with non-empty boundary in terms of their Seifert data, up to a multiplicative constant. We also show that all singular fibres of such a Seifert fibered space (aside from those of multiplicity two) can be made simplicial in the 79th barycentric subdivision of any triangulation of it.
The triangulation complexity of fibred 3-manifolds
本文主要研究在圆上纤维化的闭可定向双曲三维流形的三角剖分复杂度，证明了其与monodromy作用在mapping class group上的平移长度等价，误差仅依赖于纤维亏格的有界因子。
In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of 3-manifolds with transitive cyclic symmetry can be generalised to an infinite family of such triangulations with similarly strong combinatorial properties. In particular, we construct triangulations of Seifert fibred spaces with transitive cyclic symmetry where the symmetry preserves the fibres and acts non-trivially on the homology of the spaces. The triangul…
It is not known whether there exists a computable function bounding the number of Pachner moves needed to connect any two triangulation of a compact 3-manifold. In this paper we find an explicit bound of this kind for all Haken 3-manifolds which contain no fibred submanifolds as strongly simple pieces of their JSJ-decomposition. The explicit formula for the bound is in terms of the number of tetrahedra in the two triangulations. This implies a conceptually trivial algorithm for recognising any non-fibred knot complement among all 3-manifolds.
Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II
本文肯定回答 Kollár 的三个问题，对有理曲面上的实三维流形给出 Seifert 纤维与透镜空间的精确估计，推广 Comessatti 定理，并构造非可定向双曲型基的极小例子。
We compute the $p$-primary components of the linking pairings of orientable 3-manifolds admitting a fixed-point free $S^1$-action. Using this, we show that any non-singular linking pairing on a finite abelian group with homogeneous 2-primary summand is realized by such a manifold. However, some pairings on inhomogeneous 2-groups are not realizable by any Seifert fibred 3-manifold.
On 3-manifolds admitting co-orientable taut foliations, but none with vanishing Euler class
本文构造了允许共定向紧致叶面但无消失欧拉类的3维流形，揭示了其与L空间猜想及左可序基本群的关系。
The taut polynomial and the Alexander polynomial
本文证明转向三角剖分的绷紧多项式等于底层流形的扭结亚历山大多项式，并建立其与非扭结亚历山大多项式及 Dehn 填充后流形亚历山大多项式的关系，推广了 McMullen 定理并给出可定向纤维类存在的充要条件。
SU(2)-abelian graph manifolds with a single JSJ torus
该研究针对粘合两个带两条奇异纤维的圆盘上Seifert纤维化空间得到的单JSJ环面图流形，依据Seifert系数完成SU(2)-阿贝尔3-流形的分类，并证明这类流形是Heegaard Floer同调L-空间。
Recognition of Seifert fibered spaces with boundary is in NP
本文证明了识别带非空边界的三角剖分3-流形是否具有Seifert纤维化结构属于NP，且生成其Seifert数据属于FNP，其核心是通过证明存在小度数基本水平曲面和总权重受三角剖分大小平方指数界限制的完整正规垂直环面集合来实现。
It was recently shown that there exists an explicit bound for the number of Pachner moves needed to connect any two triangulation of any Haken 3-manifold which contains no fibred sub-manifolds as strongly simple pieces of its JSJ-decomposition. In this paper we prove a generalisation of that result to all knot complements. The explicit formula for the bound is in terms of the numbers of tetrahedra in the two triangulations. This gives a conceptually trivial algorithm for recognising any knot complement among all 3-manifolds.
We show that for a special alternating link diagram, the following three polynomials are essentially the same: a) the part of the HOMFLY polynomial that corresponds to the leading term in the Alexander polynomial; b) the $h$-vector for a triangulation of the root polytope of the Seifert graph and c) the enumerator of parking functions for the planar dual of the Seifert graph. These observations yield formulas for the maximal $z$-degree part of the HOMFLY polynomial of an arbitrary homogeneous link as well. Our result is part of a program aimed at reading HOMFLY coefficients out of Heegaard Flo…
Given a triangulation of a closed, oriented, irreducible, atoroidal 3-manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded by non-negative integer weights, 14 for each 3-simplex, that describe how many copies of each oriented normal disc type there are. The Euler characteristic and homology class are both linear functions of the weights. There is a convex polytope in the space of weights, defined by linear equations given by the combinatorics of the triangulation, whose image under the…
We calculate Blanchfield pairings of 3-manifolds. In particular, we give a formula for the Blanchfield pairing of a fibred 3-manifold and we give a new proof that the Blanchfield pairing of a knot can be expressed in terms of a Seifert matrix.
We show that $\mathbb{S}ol^3\times\mathbb{E}^1$-manifolds are Seifert fibred, with general fibre the torus, and base one of the seven flat 2-orbifolds $T, Kb, \mathbb{A}, \mathbb{M}b, S(2,2,2,2), P(2,2)$ or $\mathbb{D}(2,2)$, and outline a classification of such 4-manifolds.
Motivated by fixed-parameter tractable (FPT) problems in computational topology, we consider the treewidth of a compact, connected 3-manifold $M$ defined by \[ \operatorname{tw}(M) = \min\{\operatorname{tw}(Γ(\mathcal{T})):\mathcal{T}~\text{is a triangulation of }M\}, \] where $Γ(\mathcal{T})$ denotes the dual graph of $\mathcal{T}$. In this setting the relationship between the topology of a 3-manifold and its treewidth is of particular interest. First, as a corollary of work of Jaco and Rubinstein, we prove that for any closed, orientable 3-manifold $M$ the treewidth $\operatorname{tw}(M)$ is…
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