Seifert Form Research Articles

Slicing knots in definite 4-manifolds

We study the C P 2 \mathbb {CP}^2 -slicing number of knots, i.e. the smallest m ≥ 0 m\geq 0 such that a knot K ⊆ S 3 K\subseteq S^3 bounds a properly embedded, null-homologous disk in a punctured connected sum ( # m C P 2 ) × (\#^m\mathbb {CP}^2)^{\times } . We find knots for which the smooth and topological C P 2 \mathbb {CP}^2 -slicing numbers are both finite, nonzero, and distinct. To do this, we give a lower bound on the smooth C P 2 \mathbb {CP}^2 -slicing number of a knot in terms of its double branched cover and an upper bound on the topological C P 2 \mathbb {CP}^2 -slicing number in terms of the Seifert form.

A note on simple K3 singularities and families of weighted K3 surfaces

We discuss properties of the Seifert form for simple K3 singularities, and of the Picard lattices of families of weighted K3 surfaces. We study a collection $$\mathcal {M}_{(\rho ,\,\delta )}$$ of K3 surfaces polarized by their Picard lattices that are in the set $$\mathcal {L}_{(\rho ,\,\delta )}$$ of certain lattices. We also report a numerical formula that relates the Seifert form for the singularities and the Picard lattice of the family.

Seifert form of chain-type invertible singularities

In this paper, we confirm a conjecture of Orlik and Randell from 1977 on the Seifert form of chain-type invertible singularities. We use Lefschetz bifibration techniques as developed by Seidel (inspired by Arnold and Donaldson) and take advantage of the symmetries at hand. We believe that our method will be useful in understanding the homological/categorical version of Berglund–Hübsch mirror conjecture for invertible singularities.

On a quadratic form associated with a surface automorphism and its applications to Singularity Theory

We study the nilpotent part N′ of a pseudo-periodic automorphism h of a real oriented surface with boundary Σ. We associate a quadratic form Q defined on the first homology group (relative to the boundary) of the surface Σ. Using the twist formula and techniques from mapping class group theory, we prove that the form Q̃ obtained after killing kerN is positive definite if all the screw numbers associated with certain orbits of annuli are positive. We also prove that the restriction of Q̃ to the absolute homology group of Σ is even whenever the quotient of the Nielsen–Thurston graph under the action of the automorphism is a tree. The case of monodromy automorphisms of Milnor fibers Σ=F of germs of curves on normal surface singularities is discussed in detail, and the aforementioned results are specialized to such situation. This form Q is determined by the Seifert form but can be much more easily computed. Moreover, the form Q̃ is computable in terms of the dual resolution or semistable reduction graph, as illustrated with several examples. Numerical invariants associated with Q̃ are able to distinguish plane curve singularities with different topological types but same spectral pairs. Finally, we discuss a generic linear germ defined on a superisolated surface. In this case the plumbing graph is not a tree and the restriction of Q̃ to the absolute monodromy of Σ=F is not even.

Baskets and fibred links realizing An

We prove that basket links, whose symmetrized Seifert form is congruent to the Cartan matrix of the simply laced Dynkin diagram An, 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 An, that are not isotopic to T(2,n+1).

Maximally-graded matrix factorizations for an invertible polynomial of chain type

In 1977, Orlik–Randell construct a nice integral basis of the middle homology group of the Milnor fiber associated to an invertible polynomial of chain type and they conjectured that it is represented by a distinguished basis of vanishing cycles.The purpose of this paper is to prove the algebraic counterpart of the Orlik–Randell conjecture. Under the homological mirror symmetry, we may expect that the triangulated category of maximally-graded matrix factorizations for the Berglund–Hübsch transposed polynomial admits a full exceptional collection with a nice numerical property. Indeed, we show that the category admits a Lefschetz decomposition with respect to a polarization in the sense of Kuznetsov–Smirnov, whose Euler matrix are calculated in terms of the “zeta function” of the inverse of the polarization.As a corollary, it turns out that the homological mirror symmetry holds at the level of the Grothendieck group in the following sense; the Grothendieck group of the category with the Euler form is isomorphic to the middle homology group with the (quasi-)Seifert form (with a suitable sign).

On families of fibred knots with equal Seifert forms

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.

On definite strongly quasipositive links and L-space branched covers

We investigate the problem of characterising the family of strongly quasipositive links which have definite symmetrised Seifert forms and apply our results to the problem of determining when such a link can have an L-space cyclic branched cover. In particular, we show that if δn=σ1σ2…σn−1 is the dual Garside element and b=δnkP∈Bn is a strongly quasipositive braid whose braid closure bˆ is definite, then k≥2 implies that bˆ is one of the torus links T(2,q),T(3,4),T(3,5) or pretzel links P(−2,2,m),P(−2,3,4). Applying [7, Theorem 1.1] we deduce that if one of the standard cyclic branched covers of bˆ is an L-space, then bˆ is one of these links. We show by example that there are strongly quasipositive braids δnP whose closures are definite but not one of these torus or pretzel links. We also determine the family of definite strongly quasipositive 3-braids and show that their closures coincide with the family of strongly quasipositive 3-braids with an L-space branched cover.

On classical upper bounds for slice genera

We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fundamental group. We characterize the algebraic genus in terms of cobordisms in three-space, and explore the connections to other knot invariants related to the Seifert form, the Blanchfield form, knot genera and unknotting. Employing Casson-Gordon invariants, we discuss the algebraic genus as a candidate for the optimal upper bound for the topological slice genus that is determined by the S-equivalence class of Seifert matrices.

Real Seifert Forms and Polarizing Forms of Steenbrink Mixed Hodge Structures

An isolated hypersurface singularity comes equipped with many different pairings on different spaces, the intersection form and the Seifert form on the Milnor lattice, a polarizing form for a mixed Hodge structure on a dual space, and a flat pairing on the cohomology bundle. This paper describes them and their relations systematically in an abstract setting. We expect applications also in other areas than singularity theory. A good part of the paper is elementary, but not well known: the classification of irreducible Seifert form pairs, the polarizing form on the generalized eigenspace with eigenvalue 1, an automorphism from a Fourier–Laplace transformation which involves the Gamma function and which relates Seifert form and polarizing form and a flat pairing on the cohomology bundle. New is a correction of a Thom–Sebastiani formula for Steenbrink’s Hodge filtration in the case of singularities. It uses the Fourier–Laplace transformation. A special case is a square root of a Tate twist for Steenbrink mixed Hodge structures.

\mu-constant monodromy groups and Torelli results for the quadrangle singularities and the bimodal series

This paper is a sequel to [He11] and [GH17]. In [He11] a notion of marking of isolated hypersurface singularities was defined, and a moduli space $M_\mu^{mar}$ for marked singularities in one $\mu$-homotopy class of isolated hypersurface singularities was established. It is an analogue of a Teichm\uller space. It comes together with a $\mu$-constant monodromy group $G^{mar}\subset G_{\mathbb{Z}}$. Here $G_{\mathbb{Z}}$ is the group of automorphisms of a Milnor lattice which respect the Seifert form. It was conjectured that $M_\mu^{mar}$ is connected. This is equivalent to $G^{mar}= G_{\mathbb{Z}}$. Also Torelli type conjectures were formulated. In [He11] and [GH17] $M_\mu^{mar}, G_{\mathbb{Z}}$ and $G^{mar}$ were determined and all conjectures were proved for the simple, the unimodal and the exceptional bimodal singularities. In this paper the quadrangle singularities and the bimodal series are treated. The Torelli type conjectures are true. But the conjecture $G^{mar}= G_{\mathbb{Z}}$ and $M_\mu^{mar}$ connected does not hold for certain subseries of the bimodal series.

Knots having the same Seifert form and primary decomposition of knot concordance

We show that for each Seifert form of an algebraically slice knot with nontrivial Alexander polynomial, there exists an infinite family of knots having the Seifert form such that the knots are linearly independent in the knot concordance group and not concordant to any knot with coprime Alexander polynomial. Key ingredients for the proof are Cheeger–Gromov–von Neumann [Formula: see text]-invariants for amenable groups developed by Cha–Orr and polynomial splittings of metabelian [Formula: see text]-invariants.

On Calculating the Slice Genera of 11- and 12-Crossing Knots

ABSTRACTThis article contains the results of efforts to determine the values of the smooth and the topological slice genus of 11- and 12-crossing knots. Upper bounds for these genera were produced by using a computer to search for genus one concordances between knots. For the topological slice genus, further upper bounds were produced using the algebraic genus. Lower bounds were obtained using a new obstruction from the Seifert form and by the use of Donaldson’s diagonalization theorem. These results complete the computation of the topological slice genera for all knots with at most 11 crossings and leaves the smooth genera unknown for only two 11-crossing knots. For 12 crossings, there remain merely 25 knots whose smooth or topological slice genus is unknown.

Orbifold Milnor lattice and orbifold intersection form

For a germ of a quasihomogeneous function with an isolated critical point at the origin invariant with respect to an appropriate action of a finite abelian group, H. Fan, T. Jarvis, and Y. Ruan defined the so-called quantum cohomology group. It is considered as the main object of the quantum singularity theory (FJRW-theory). We define orbifold versions of the monodromy operator on the quantum (co)homology group, of the Milnor lattice, of the Seifert form and of the intersection form. We also describe some symmetry properties of invariants of invertible polynomials refining the known ones.

Double L–groups and doubly slice knots

We develop a theory of chain complex double-cobordism for chain complexes equipped with Poincar\'{e} duality. The resulting double-cobordism groups are a refinement of Ranicki's torsion algebraic $L$-groups for localisations of a commutative ring with involution. The refinement is analogous to the difference between metabolic and hyperbolic linking forms. We apply the double $L$-groups in high-dimensional knot theory to define an invariant for doubly-slice $n$-knots. We prove that the "stably doubly-slice implies doubly-slice" property holds (algebraically) for Blanchfield forms, Seifert forms and for the Blanchfield complexes of $n$-knots for $n\geq 1$.

The Alexander module, Seifert forms, and categorification

We show that bordered Floer homology provides a categorification of a TQFT described by Donaldson. This, in turn, leads to a proof that both the Alexander module of a knot and the Seifert form are completely determined by Heegaard Floer theory.

On the stable $4$-genus of knots with indefinite Seifert form

Under a simple assumption on Seifert surfaces, we characterise knots whose stable topological 4-genus coincides with the genus.

On the semicontinuity of the mod 2 spectrum of hypersurface singularities

We use purely topological methods to prove the semicontinuity of the mod 2 \mod 2 spectrum of local isolated hypersurface singularities in C n + 1 \mathbb {C}^{n+1} , using Seifert forms of high-dimensional non-spherical links, the Levine–Tristram signatures and the generalized Murasugi–Kawauchi inequality.

Smooth slice boundary links whose derivative links have nonvanishing Milnor invariants

We give an example of a 3-component smoothly slice boundary link, each of whose components has a genus one Seifert surface, such that any metaboliser of the boundary link Seifert form is represented by 3 curves on the Seifert surfaces that form a link with nonvanishing Milnor triple linking number. We also give a generalisation to m-component links and higher Milnor invariants. We prove that our examples are ribbon and that all ribbon links are boundary slice.

Positive braids of maximal signature

We characterise positive braid links with positive Seifert form via a finite number of forbidden minors. From this we deduce a one-to-one correspondence between prime positive braid links of maximal signature and simply laced Dynkin diagrams, as well as a simple classification of alternating positive braid knots.