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Stefan Friedl

Publications and source records attributed to Stefan Friedl.

At least 73 records · Page 4Linked to original sources
arXiv2014

Three flavors of twisted invariants of knots

The Alexander polynomial of a knot has been generalized in three different ways to give twisted invariants. The resulting invariants are usually referred to as twisted Alexander polynomials, higher-order Alexander polynomials and $L^2$-Alexander invariants of knots. We quickly recall the definitions and we summarize and compare some of their properties. We also report on work by the authors on $L^2$-Alexander torsions and we conclude the paper with several conjectures on $L^2$-Alexander torsions.

math.GT↗
arXiv2014

Twist spinning of knots and metabolizers of Blanchfield pairings

In a classic paper Zeeman introduced the k-twist spin of a knot K and showed that the exterior of a twist spin fibers over S^1. In particular this result shows that the knot K # -K is doubly slice. In this paper we give a quick proof of Zeeman's result. The k-twist spin of K also gives rise to two metabolizers for K # -K and we determine these two metabolizers precisely.

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arXiv2014

A specious unlinking strategy

We show that the following unlinking strategy does not always yield an optimal sequence of crossing changes: first split the link with the minimal number of crossing changes, and then unknot the resulting components.

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arXiv2014

Blanchfield forms and Gordian distance

Given a link in $S^3$ we will use invariants derived from the Alexander module and the Blanchfield pairing to obtain lower bounds on the Gordian distance between links, the unlinking number and various splitting numbers. These lower bounds generalise results recently obtained by Kawauchi. We give an application restricting the knot types which can arise from a sequence of splitting operations on a link. This allows us to answer a question asked by Colin Adams in 1996.

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arXiv2014

Sutured Floer homology, fibrations, and taut depth one foliations

For an oriented irreducible 3-manifold M with non-empty toroidal boundary, we describe how sutured Floer homology ($SFH$) can be used to determine all fibered classes in $H^1(M)$. Furthermore, we show that the $SFH$ of a balanced sutured manifold $(M,γ)$ detects which classes in $H^1(M)$ admit a taut depth one foliation such that the only compact leaves are the components of $R(γ)$. The latter had been proved earlier by the first author under the extra assumption that $H_2(M)=0$. The main technical result is that we can obtain an extremal $\text{Spin}^c$-structure $\mathfrak{s}$ (i.e., one that is in a `corner' of the support of $SFH$) via a nice and taut sutured manifold decomposition even when $H_2(M) \neq 0$, assuming the corresponding group $SFH(M,γ,\mathfrak{s})$ has non-trivial Euler characteristic.

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arXiv2014

Virtually symplectic fibered 4-manifolds

We mostly determine which closed smooth oriented 4-manifolds fibering over lower dimensional manifolds are virtually symplectic, i.e. finitely covered by symplectic 4-manifolds.

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arXiv2013

Thompson's group F is not SCY

In this note we prove that Thompson's group F cannot be the fundamental group of a symplectic 4-manifold with trivial canonical class by showing that its Hausmann-Weinberger invariant q(F) is strictly positive.

math.GT↗
arXiv2013

The virtual fibering theorem for 3-manifolds

In 2007 Agol showed that if N is an aspherical compact 3-manifold with empty or toroidal boundary such that its fundamental group is virtually RFRS, then $N$ is virtually fibered. We give a largely self-contained proof of Agol's theorem using complexities of sutured manifolds.

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arXiv2013

Rank gradients of infinite cyclic covers of 3-manifolds

Given a 3-manifold M with no spherical boundary components, and a primitive class ϕin H^1(M;Z), we show that the following are equivalent: (1) ϕis a fibered class, (2) the rank gradient of (M,ϕ) is zero, (3) the Heegaard gradient of (M,ϕ) is zero.

math.GT↗
arXiv2013

Splittings of knot groups

Let K be a knot of genus g. If K is fibered, then it is well known that the knot group pi(K) splits only over a free group of rank 2g. We show that if K is not fibered, then pi(K) splits over non-free groups of arbitrarily large rank. Furthermore, if K is not fibered, then pi(K) splits over every free group of rank at least 2g. However, pi(K) cannot split over a group of rank less than 2g. The last statement is proved using the recent results of Agol, Przytycki-Wise and Wise.

math.GT↗
arXiv2013

On the algebraic unknotting number

The algebraic unknotting number u_a(K) of a knot K was introduced by Hitoshi Murakami. It equals the minimal number of crossing changes needed to turn K into an Alexander polynomial one knot. In a previous paper the authors used the Blanchfield form of a knot K to define an invariant n(K) and proved that n(K) is a lower bound on u_a(K). They also showed that n(K) subsumes all previous classical lower bounds on the (algebraic) unknotting number. In this paper we prove that n(K)=u_a(K).

math.GT↗
arXiv2013

Splitting numbers of links

The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the splitting numbers of links with 9 or fewer crossings. Also, with these techniques, we either reprove or improve upon the lower bounds for splitting numbers of links computed by J. Batson and C. Seed using Khovanov homology.

math.GT↗
arXiv2013

The unknotting number and classical invariants II

In [BF12] the authors associated to a knot K an invariant n_R(K) which is defined using the Blanchfield form and which gives a lower bound on the unknotting number. In this paper we express n_R(K) in terms of Levine-Tristram signatures and nullities of K. In the proof we also show that the Blanchfield form with real coefficients is diagonalizable.

math.GT↗
arXiv2013

Twisted Alexander invariants detect trivial links

It follows from earlier work of Silver-Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexander modules detect trivial links.

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arXiv2013

3-manifold groups

We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

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arXiv2013

A note on the growth of Betti numbers and ranks of 3-manifold groups

Let N be an irreducible, compact 3-manifold with empty or toroidal boundary which is not a closed graph manifold. Using recent work of Agol, Kahn-Markovic and Przytycki-Wise we will show that pi_1(N) admits a cofinal filtration with `fast' growth of Betti numbers as well as a cofinal filtration of pi_1(N) with `slow' growth of ranks.

math.GT↗