## Thesis etd-09092010-162353 |

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Thesis type

Tesi di dottorato di ricerca

Author

GARCIA LECUONA, ANA

URN

etd-09092010-162353

Thesis title

On the slice-ribbon conjecture for Montesinos knots

Academic discipline

MAT/03

Course of study

MATEMATICA

Supervisors

**tutor**Prof. Lisca, Paolo

Keywords

- montesinos knots
- rational homology balls
- slice-ribbon conjecture

Graduation session start date

15/09/2010

Availability

Withheld

Release date

15/09/2050

Summary

A celebrated open problem in knot theory, posed by Fox [Fo] in 1962, is the so-called slice-ribbon conjecture: every (smoothly) slice knot in the 3-sphere is ribbon. The concepts of being slice and ribbon are basic in knot theory: a knot is slice if it is the boundary of a (smoothly) embedded disc in the 4-ball, and it is ribbon if it is slice and admits a slicing disc without local maxima for the radius function in the 4-ball. This conjecture is the main subject of this thesis.

The problem of detecting slice knots is very deep, with distinct differences between the classical case of the embeddings of S^1 in S^3 and the generalizations to higher dimensional knots. In fact, Kervaire showed that all even

%Buscar la referencia a Kervaire a Levine, a Casson y Gordon (1978)

dimensional knots are slice [Ke], and for odd dimensional knots of dimension higher than one Levine [Lev,Lev2] found a computable algebraic method for determining whether or not a knot is slice. Perhaps, the most significant work on the classical slice problem, definitively discriminating it from higher dimensions, is due to Casson and Gordon in 1978 [CG], but their method is in general very difficult to apply.

At the time of writing this thesis, the validity of Fox's slice-ribbon conjecture is far from being known. However, in recent work, Lisca [Li] has established it for the case of 2-bridge knots, and Greene and Jabuka [GJ] for 3-stranded pretzel knots P(p,q,r) with p,q,r odd. Both families of knots belong to the significantly broader class of Montesinos knots, first defined in [Mo2]. In this thesis we prove the validity of the slice-ribbon conjecture for a subfamily of Montesinos knots, the family P defined and motivated in Section I.3. Moreover, we give a necessary condition for the sliceness of pretzel knots with an arbitrary number of strands and at least one even parameter. This condition is also sufficient in the case of 3-stranded pretzel knots P(p,q,r) with exactly one among p,q,r even (not considered in [GJ]). We further show that the slice-ribbon conjecture holds for this family. Note that, this result together with [GJ], show that the slice-ribbon conjecture holds for all 3-stranded pretzel knots. Other results concerning sliceness for Montesinos knots can be found in [Wi], where Williams shows that no member of a certain five parameter family of Montesinos knots is slice.

A brief synopsis of our method of proof, which follows in part the approach of [Li,GJ], goes as follows: in a first step, given a Montesinos knot K we consider a negative-definite 4-plumbing M whose boundary Y is the double cover of S^3 branched over K. For Montesinos knots the 3-manifold Y is always a Seifert space. If K is slice then Y also bounds a rational homology ball. Gluing this ball to M along their common boundary yields a smooth, closed, negative definite 4-manifold X. As such, according to Donaldson's theorem [Do], its intersection form has to be standard. This obstruction suffices to determine all slice knots among 2-bridge knots [Li] and among the Montesinos knots in family P in our case. In contrast, when studying pretzel knots this obstruction is not enough to determine all the slice ones. In [GJ] the authors succeed using the above obstruction supplemented by Heegaard Floer homology techniques. We apply the same Heegaard Floer homology techniques plus Rasmussen's s-invariant to determine all slice 3-stranded pretzel knots P(p,q,r) with exactly one even parameter.

This thesis can be conceptually divided into two parts. The first one consists of the first three chapters and is devoted to the study of the slice-ribbon conjecture for the family of Montesinos knots P. In Chapter I we give a quick overview of Montesinos links, Seifert spaces and plumbings and we explain in full detail how to use Donaldson's theorem as an obstruction to sliceness for Montesinos knots. Moreover, we define the object of our study, the family P, state the main result of this part of the thesis, and introduce the necessary notation and definitions in order to present the complete list, Theorems I.9 and I.10, of Seifert spaces associated to Montesinos links in P that are the boundary of a 4-dimensional plumbing whose intersection lattice admits an embedding into the standard negative diagonal intersection lattice. We postpone the long and technical proof of Theorems I.9 and I.10 to Chapter III. The idea of the proof can be briefly described as follows. Montesinos links in P are described by means of a star-shaped graph whose vertices carry an integer weight. In Chapter III we introduce an operation, the contraction, which applied to a weighted graph returns another graph with one vertex less. Under certain assumptions this operation can be iterated until we obtain a "minimal" graph. A thorough algebro-combinatorial study of these contractions leads to the proof of Theorems I.9 and I.10.

In contrast, Chapter II has a more geometric flavour. Theorems I.9 and I.10 give a complete list of possibly slice knots inside P. In Chapter II we show that all of them are actually ribbon. In this way we establish the validity of the slice-ribbon conjecture for the Montesinos knots belonging to the family P. The approach we use in this chapter differs from the one followed in [Li,GJ]. Seifert spaces are the boundary of 4-dimensional plumbings admitting a Kirby diagram which is strongly invertible with respect to an involution $u$ of S^3. This involution turns the Seifert space into the double cover of S^3 branched over a Montesinos link. In Chapter II we show how to perform a surgery on Y in order to obtain S^1xS^2. This surgery can be understood by adding a 2-handle to the Kirby diagram associated to Y and taking the boundary. The key point is that the addition of the 2-handle can be done in such a way that one obtains again a strongly invertible link with respect to u. This implies, by a theorem due to Montesinos, that the addition of the 2-handle corresponds to a ribbon move on the Montesinos link which turns it into two disjoint unknots. It follows that if the Montesinos link is a knot, then it is ribbon. At the end of Chapter II we prove Theorem I.6, which is the main result of this first part of the thesis.

Chapter IV is a collection of minor results obtained as a byproduct of the analysis of the family P. In it we consider other families of Montesinos links, which are slight modifications of P, and we highlight the differences between P and its modifications.

The second part of the thesis, Chapter V, deals with the slice-ribbon conjecture for pretzel knots P(p_1,...,p_n) having at least one even parameter. We start by proposing a conjecture giving a criterion to determine whether a pretzel knot is ribbon. We prove it in the case of three-stranded pretzel knots P(p_1,p_2,p_3) (for the case p_1,p_2,p_3 odd it follows from [GJ,Theorem 1.1] and we also show that the slice-ribbon conjecture holds in this case. We develop a further analysis on the sliceness of pretzel knots with an arbitrary number of strands and at least one even parameter, which supports our conjecture. Some of the conclusions are drawn from studying the double branched cover of pretzel knots, the Seifert spaces Y(p_1,...,p_n). These spaces are invariant under any permutation of the parameters p_1,...,p_n, while the corresponding pretzel knots are only invariant under cyclic permutations, order reversal permutations, and compositions of these. Therefore the statement of our main result concerning the sliceness of pretzel knots is given only up to mutation. First, we study a combination of two obstructions to sliceness: the vanishing of the knot signature and the diagonalization obstruction used in the preceding chapters. For some subfamilies of pretzel knots the study of these two obstructions is enough to prove the desired result. However, for some other cases, we must also consider obstructions coming from Heegaard Floer homology, as well as Rasmussen's s-invariant.

REFERENCES

[Do] S.K. Donaldson, The orientation of Yang-Mills moduli spaces and 4-manifold topology, J. Differential Geom., 26(3), 1987, 397--428.

[CG] A.J. Casson, C.McA. Gordon, On slice knots in dimension three, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., 2, 1976, 39--53.

[CH] A.J. Casson, J.L. Harer, Some homology lens spaces which bound rational homology balls, Pacific J. Math., 96(1), 1981, 23--36.

[Fo] R. H. Fox, Some problems in knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute), (1962), 168--176.

[GJ] J. Greene, S. Jabuka, The slice-ribbon conjecture for $3$-stranded pretzel knots, 2007, arXiv:0706.3398v2

[Ke] M.A. Kervaire, Les nœuds de dimensions supérieures, Bull. Soc. Math. France, 93, 1965, 225--271.

[Lev] J. Levine, Knot cobordism groups n codimension two, Comment. Math. Helv., 44, 1969, 229--244.

[Lev2] J. Levine, Invariants of knot cobordism, Invent. Math., 8, 1969, 98--110.

[Li] P. Lisca, Lens spaces, rational balls and the ribbon conjecture, Geom. Top., 11, 2007, 429--472.

[Mo] J.M. Montesinos, Variedades de Seifert que son recubridores ciclicos ramificados de dos hojas, Bol. Soc. Mat. Mexicana (2), 18, 1973, 1--32.

[Wi] L. Williams, Obstructing sliceness in a family of Montesinos knots, 2008, arXiv:0809.1247

The problem of detecting slice knots is very deep, with distinct differences between the classical case of the embeddings of S^1 in S^3 and the generalizations to higher dimensional knots. In fact, Kervaire showed that all even

%Buscar la referencia a Kervaire a Levine, a Casson y Gordon (1978)

dimensional knots are slice [Ke], and for odd dimensional knots of dimension higher than one Levine [Lev,Lev2] found a computable algebraic method for determining whether or not a knot is slice. Perhaps, the most significant work on the classical slice problem, definitively discriminating it from higher dimensions, is due to Casson and Gordon in 1978 [CG], but their method is in general very difficult to apply.

At the time of writing this thesis, the validity of Fox's slice-ribbon conjecture is far from being known. However, in recent work, Lisca [Li] has established it for the case of 2-bridge knots, and Greene and Jabuka [GJ] for 3-stranded pretzel knots P(p,q,r) with p,q,r odd. Both families of knots belong to the significantly broader class of Montesinos knots, first defined in [Mo2]. In this thesis we prove the validity of the slice-ribbon conjecture for a subfamily of Montesinos knots, the family P defined and motivated in Section I.3. Moreover, we give a necessary condition for the sliceness of pretzel knots with an arbitrary number of strands and at least one even parameter. This condition is also sufficient in the case of 3-stranded pretzel knots P(p,q,r) with exactly one among p,q,r even (not considered in [GJ]). We further show that the slice-ribbon conjecture holds for this family. Note that, this result together with [GJ], show that the slice-ribbon conjecture holds for all 3-stranded pretzel knots. Other results concerning sliceness for Montesinos knots can be found in [Wi], where Williams shows that no member of a certain five parameter family of Montesinos knots is slice.

A brief synopsis of our method of proof, which follows in part the approach of [Li,GJ], goes as follows: in a first step, given a Montesinos knot K we consider a negative-definite 4-plumbing M whose boundary Y is the double cover of S^3 branched over K. For Montesinos knots the 3-manifold Y is always a Seifert space. If K is slice then Y also bounds a rational homology ball. Gluing this ball to M along their common boundary yields a smooth, closed, negative definite 4-manifold X. As such, according to Donaldson's theorem [Do], its intersection form has to be standard. This obstruction suffices to determine all slice knots among 2-bridge knots [Li] and among the Montesinos knots in family P in our case. In contrast, when studying pretzel knots this obstruction is not enough to determine all the slice ones. In [GJ] the authors succeed using the above obstruction supplemented by Heegaard Floer homology techniques. We apply the same Heegaard Floer homology techniques plus Rasmussen's s-invariant to determine all slice 3-stranded pretzel knots P(p,q,r) with exactly one even parameter.

This thesis can be conceptually divided into two parts. The first one consists of the first three chapters and is devoted to the study of the slice-ribbon conjecture for the family of Montesinos knots P. In Chapter I we give a quick overview of Montesinos links, Seifert spaces and plumbings and we explain in full detail how to use Donaldson's theorem as an obstruction to sliceness for Montesinos knots. Moreover, we define the object of our study, the family P, state the main result of this part of the thesis, and introduce the necessary notation and definitions in order to present the complete list, Theorems I.9 and I.10, of Seifert spaces associated to Montesinos links in P that are the boundary of a 4-dimensional plumbing whose intersection lattice admits an embedding into the standard negative diagonal intersection lattice. We postpone the long and technical proof of Theorems I.9 and I.10 to Chapter III. The idea of the proof can be briefly described as follows. Montesinos links in P are described by means of a star-shaped graph whose vertices carry an integer weight. In Chapter III we introduce an operation, the contraction, which applied to a weighted graph returns another graph with one vertex less. Under certain assumptions this operation can be iterated until we obtain a "minimal" graph. A thorough algebro-combinatorial study of these contractions leads to the proof of Theorems I.9 and I.10.

In contrast, Chapter II has a more geometric flavour. Theorems I.9 and I.10 give a complete list of possibly slice knots inside P. In Chapter II we show that all of them are actually ribbon. In this way we establish the validity of the slice-ribbon conjecture for the Montesinos knots belonging to the family P. The approach we use in this chapter differs from the one followed in [Li,GJ]. Seifert spaces are the boundary of 4-dimensional plumbings admitting a Kirby diagram which is strongly invertible with respect to an involution $u$ of S^3. This involution turns the Seifert space into the double cover of S^3 branched over a Montesinos link. In Chapter II we show how to perform a surgery on Y in order to obtain S^1xS^2. This surgery can be understood by adding a 2-handle to the Kirby diagram associated to Y and taking the boundary. The key point is that the addition of the 2-handle can be done in such a way that one obtains again a strongly invertible link with respect to u. This implies, by a theorem due to Montesinos, that the addition of the 2-handle corresponds to a ribbon move on the Montesinos link which turns it into two disjoint unknots. It follows that if the Montesinos link is a knot, then it is ribbon. At the end of Chapter II we prove Theorem I.6, which is the main result of this first part of the thesis.

Chapter IV is a collection of minor results obtained as a byproduct of the analysis of the family P. In it we consider other families of Montesinos links, which are slight modifications of P, and we highlight the differences between P and its modifications.

The second part of the thesis, Chapter V, deals with the slice-ribbon conjecture for pretzel knots P(p_1,...,p_n) having at least one even parameter. We start by proposing a conjecture giving a criterion to determine whether a pretzel knot is ribbon. We prove it in the case of three-stranded pretzel knots P(p_1,p_2,p_3) (for the case p_1,p_2,p_3 odd it follows from [GJ,Theorem 1.1] and we also show that the slice-ribbon conjecture holds in this case. We develop a further analysis on the sliceness of pretzel knots with an arbitrary number of strands and at least one even parameter, which supports our conjecture. Some of the conclusions are drawn from studying the double branched cover of pretzel knots, the Seifert spaces Y(p_1,...,p_n). These spaces are invariant under any permutation of the parameters p_1,...,p_n, while the corresponding pretzel knots are only invariant under cyclic permutations, order reversal permutations, and compositions of these. Therefore the statement of our main result concerning the sliceness of pretzel knots is given only up to mutation. First, we study a combination of two obstructions to sliceness: the vanishing of the knot signature and the diagonalization obstruction used in the preceding chapters. For some subfamilies of pretzel knots the study of these two obstructions is enough to prove the desired result. However, for some other cases, we must also consider obstructions coming from Heegaard Floer homology, as well as Rasmussen's s-invariant.

REFERENCES

[Do] S.K. Donaldson, The orientation of Yang-Mills moduli spaces and 4-manifold topology, J. Differential Geom., 26(3), 1987, 397--428.

[CG] A.J. Casson, C.McA. Gordon, On slice knots in dimension three, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., 2, 1976, 39--53.

[CH] A.J. Casson, J.L. Harer, Some homology lens spaces which bound rational homology balls, Pacific J. Math., 96(1), 1981, 23--36.

[Fo] R. H. Fox, Some problems in knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute), (1962), 168--176.

[GJ] J. Greene, S. Jabuka, The slice-ribbon conjecture for $3$-stranded pretzel knots, 2007, arXiv:0706.3398v2

[Ke] M.A. Kervaire, Les nœuds de dimensions supérieures, Bull. Soc. Math. France, 93, 1965, 225--271.

[Lev] J. Levine, Knot cobordism groups n codimension two, Comment. Math. Helv., 44, 1969, 229--244.

[Lev2] J. Levine, Invariants of knot cobordism, Invent. Math., 8, 1969, 98--110.

[Li] P. Lisca, Lens spaces, rational balls and the ribbon conjecture, Geom. Top., 11, 2007, 429--472.

[Mo] J.M. Montesinos, Variedades de Seifert que son recubridores ciclicos ramificados de dos hojas, Bol. Soc. Mat. Mexicana (2), 18, 1973, 1--32.

[Wi] L. Williams, Obstructing sliceness in a family of Montesinos knots, 2008, arXiv:0809.1247

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