arXiv subjects

Stefan Friedl

Publications and source records attributed to Stefan Friedl.

125 records · Page 7Linked to original sources

arXiv2008

Metabelian SL(n,C) representations of knot groups

We give a classification of irreducible metabelian representations from a knot group into SL(n,C) and GL(n,C). If the homology of the n-fold branched cover of the knot is finite, we show that every irreducible metabelian SL(n,C) representation is conjugate to a unitary representation and that the set of conjugacy classes of such representations is finite. In that case, we give a formula for this number in terms of the Alexander polynomial of the knot. These results are the higher rank generalizations of a result of Nagasato, who recently studied irreducible, metabelian SL(2,C) representations of knot groups. Finally we deduce the existence irreducible metabelian SL(n,C) representations of the knot group for any knot with nontrivial Alexander polynomial.

math.GT↗

arXiv2007

New Constructions of Slice Links

We use techniques of Freedman and Teichner to prove that, under certain circumstances, the multi-infection of a slice link is again slice (not necessarily smoothly slice). We provide a general context for proving links are slice that includes many of the previously known results.

math.GT↗

arXiv2007

Twisted Alexander polynomials and symplectic structures

Let N be a closed, oriented 3-manifold. A folklore conjecture states that S^1 x N admits a symplectic structure only if N admits a fibration over the circle. The purpose of this paper is to provide evidence to this conjecture studying suitable twisted Alexander polynomials of N, and showing that their behavior is the same as of those of fibered 3-manifolds. In particular, we will obtain new obstructions to the existence of symplectic structures and to the existence of symplectic forms representing certain cohomology classes of S^1 x N. As an application of these results we will show that S^1 x N(P) does not admit a symplectic structure, where N(P) is the 0-surgery along the pretzel knot P = (5,-3,5), answering a question of Peter Kronheimer.

math.GT↗

arXiv2007

Symplectic S^1 x N^3, subgroup separability, and vanishing Thurston norm

Let N be a closed, oriented 3-manifold. A folklore conjecture states that $S^{1} \times N$ admits a symplectic structure if and only if $N$ admits a fibration over the circle. We will prove this conjecture in the case when N is irreducible and its fundamental group satisfies appropriate subgroup separability conditions. This statement includes 3-manifolds with vanishing Thurston norm, graph manifolds and 3-manifolds with surface subgroup separability (a condition satisfied conjecturally by all hyperbolic 3-manifolds). Our result covers, in particular, the case of 0-framed surgeries along knots of genus one. The statement follows from the proof that twisted Alexander polynomials decide fiberability for all the 3-manifolds listed above. As a corollary, it follows that twisted Alexander polynomials decide if a knot of genus one is fibered.

math.GT↗

arXiv2007

Realizations of Seifert matrices by hyperbolic knots

Recently Kearton showed that any Seifert matrix of a knot is S--equivalent to the Seifert matrix of a prime knot. We show in this note that such a matrix is in fact S--equivalent to the Seifert matrix of a hyperbolic knot. This result follows from reinterpreting this problem in terms of Blanchfield pairings and by applying results of Kawauchi.

math.GT↗

arXiv2007

Non-smoothable four-manifolds with cyclic fundamental group

In [HT], two of us constructed a closed oriented 4-dimensional manifold with fundamental group $\Z$ that does not split off $S^1\times S^3$. In this note we show that this 4-manifold, and various others derived from it, do not admit smooth structures. Moreover, we find an infinite family of 4-manifolds with exactly the same properties (and same intersection form on $H_2$). As a corollary, we obtain topologically slice knots that are not smoothly slice in any rational homology ball.

math.GT↗

arXiv2007

Nontrivial Alexander polynomials of knots and links

In this paper we present a sequence of link invariants, defined from twisted Alexander polynomials, and discuss their effectiveness in distinguish knots. In particular, we recast and extend by geometric means a recent result of Silver and Williams on the nontriviality of twisted Alexander polynomials for nontrivial knots. Furthermore building on results in [FV06b] we prove that these invariants decide if a genus one knot is fibered, and we also show that these invariants distinguish all mutants with up to 12 crossings.

math.GT↗

arXiv2006

Non-commutative Multivariable Reidemeister Torsion and the Thurston Norm

In a previous paper, the second author defined integer-valued functions delta_n on the first cohomology of a 3-manifold, generalizing McMullen's Alexander norm. It was shown that these functions give lower bounds on the Thurston norm. In this paper, we reformulate these invariants in terms of Reidemeister torsion over a non-commutative multivariable Laurent polynomial ring. This allows us to show that these functions are semi-norms.

math.GT↗

arXiv2006

Algorithm for finding boundary link Seifert matrices

We explain an algorithm for finding a boundary link Seifert matrix for a given Alexander polynomial. The algorithm depends on several choices and therefore makes it possible to find non-equivalent Seifert matrices for a given Alexander polynomial.

math.GT↗

arXiv2006

Thurston norm, fibered manifolds and twisted Alexander polynomials

Every element in the first cohomology group of a 3--manifold is dual to embedded surfaces. The Thurston norm measures the minimal `complexity' of such surfaces. For instance the Thurston norm of a knot complement determines the genus of the knot in the 3--sphere. We show that the degrees of twisted Alexander polynomials give lower bounds on the Thurston norm, generalizing work of McMullen and Turaev. Our bounds attain their most concise form when interpreted as the degrees of the Reidemeister torsion of a certain twisted chain complex. We show that these lower bounds give the correct genus bounds for all knots with 12 crossings or less, including the Conway knot and the Kinoshita--Terasaka knot which have trivial Alexander polynomial.

math.GT↗

arXiv2006

Twisted Alexander norms give lower bounds on the Thurston norm

We introduce twisted Alexander norms of a compact connected orientable 3-manifold with first Betti number bigger than one generalizing norms of McMullen and Turaev. We show that twisted Alexander norms give lower bounds on the Thurston norm of a 3-manifold. Using these we completely determine the Thurston norm of many 3-manifolds which can not be determined by norms of McMullen and Turaev.

math.GT↗

arXiv2006

Reidemeister torsion, the Thurston norm and Harvey's invariants

Recently twisted and higher order Alexander polynomials were used by Cochran, Harvey, Friedl--Kim and Turaev to give lower bounds on the Thurston norm. We first show how Reidemeister torsion relates to these Alexander polynomials. We then give lower bounds on the Thurston norm in terms of the Reidemeister torsion which contain and extend all the above lower bounds and give an elegant reformulation of the bounds of Cochran, Harvey and Turaev. The Reidemeister torsion approach also gives a natural approach to proving and extending certain monotonicity results of Cochran and Harvey.

math.GT↗

arXiv2006

The parity of the Cochran-Harvey invariants of 3-manifolds

Given a finitely presented group G and an epimorphism G to the group of integers Cochran and Harvey defined a sequence of integral invariants, which can be viewed as the degrees of higher--order Alexander polynomials. Cochran and Harvey showed that (up to a minor modification) this is a never decreasing sequence of numbers if G is the fundamental group of a 3-manifold with empty or toroidal boundary, and that these invariants give lower bounds on the Thurston norm. Using a certain Cohn localization and the duality of Reidemeister torsion we show that for a fundamental group of a 3--manifold any jump in the sequence is necessarily even. This answers in particular a question of Cochran. Furthermore using results of Turaev we show that under a mild extra hypothesis the parity of the Cochran--Harvey invariant agrees with the parity of the Thurston norm.

math.GT↗

arXiv2004

Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants

We give a useful classification of the metabelian unitary representations of pi_1(M_K), where M_K is the result of zero-surgery along a knot K in S^3. We show that certain eta invariants associated to metabelian representations pi_1(M_K) --> U(k) vanish for slice knots and that even more eta invariants vanish for ribbon knots and doubly slice knots. We show that our vanishing results contain the Casson-Gordon sliceness obstruction. In many cases eta invariants can be easily computed for satellite knots. We use this to study the relation between the eta invariant sliceness obstruction, the eta-invariant ribbonness obstruction, and the L^2-eta invariant sliceness obstruction recently introduced by Cochran, Orr and Teichner. In particular we give an example of a knot which has zero eta invariant and zero metabelian L^2-eta invariant sliceness obstruction but which is not ribbon.

math.GT↗

arXiv2003

Link concordance, boundary link concordance and eta invariants

We study the eta-invariants of links and show that in many cases they form link concordance invariants, in particular that many eta-invariants vanish for slice links. This result contains and generalizes previous invariants by Smolinsky and Cha--Ko. We give a formula for the eta-invariant for boundary links. In several intersting cases this allows us to show that a given link is not slice. We show that even more eta-invariants have to vanish for boundary slice links. We give an example of a boundary link $L$ that is not boundary slice but where all the known link concordance invariants computed so far are zero.

math.GT↗

arXiv2003

$L^2$--eta--invariants and their approximation by unitary eta--invariants

Cochran, Orr and Teichner introduced $L^2$--eta--invariants to detect highly non--trivial examples of non slice knots. Using a recent theorem by Lück and Schick we show that their metabelian $L^2$--eta--invariants can be viewed as the limit of finite dimensional unitary representations. We recall a ribbon obstruction theorem proved by the author using finite dimensional unitary eta--invariants. We show that if for a knot $K$ this ribbon obstruction vanishes then the metabelian $L^2$--eta--invariant vanishes too. The converse has been shown by the author not to be true.

math.GT↗

arXiv2003

Full signature invariants for $L_0(F(t))$

We find full invariants for detecting non--zero elements in ${L}_0(F(t))\otimes \Q$, this group plays an important role in topology in the work done by Casson and Gordon.

math.AT↗