Gauss Diagram Invariants for Knots and Links

Transcription

1 Gauss Diagram Invariants for Knots and Links

2 Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 532

3 Gauss Diagram Invariants for Knots and Links by Thomas Fiedler University of Paul Sabatier, Toulouse, France '' ~ SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

4 A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOI / Printed on acid-free paper All Rights Reserved 200 I Springer Science+ Business Media Dordrecht Originally published by Kluwer Academic Publishers in 200 I Softcover reprint of the hardcover 1st edition 200 I No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

5 to my parents Severine Alice Delphine

6 Table of Contents Preface..... Introduction and announcement. 1 The space of diagrams 1.2 The discriminant The discriminant in the case of planar surfaces 1.4 Gauss diagrams and Gauss sums Flip, orientation reversing and sensitive Gauss sums cocycles in the space of diagrams cocycles in the space of diagrams in JR cocycles in the space of diagrams in JR3 2 Invariants of knots and links by Gauss sums 2.2 The invariant of degree 1 in F 2 x IR and relations with quantum invariants The invariant of degree 2 in JR The invariants of degree 2 in F 2 x 1R 2.5 The invariant of degree 3 in JR The invariants of degree 3 for two-component links in JR3 with an unknotted component The invariants of degree 3 in F 2 x!r Some invariants with finer markings in F 2 x IR The invariants in the case of non-orientable surfaces F 2 VII IX Applications 3.2 Non-invertibility of links in JR3 3.3 Non-invertibility of knots in F 2 x IR 3.4 Chirality of knots in F 2 x!r Mutation of links in JR Orientation reversing of one component of links in JR An inequality for the Casson invariant of positive braids An inequality for the number of crossings of positive knots in JR Positivity of knots in F 2 x JR Exchange moves for diagrams in the solid torus 3.11 Alternating knots in the solid torus

7 VI 4 Global knot theory in F 2 x lr 4.0 Introduction and main results Isotopy of global knots Construction oft-invariants for global knots 4.3 T-invariants separate Z/22-pure global knots in T 2 x lr 4.4 Non-invertibility of knots in T 2 x lr A remark on quantum invariants for knots in T 2 x lr. 4.6 Non-invertibility of links in T-invariants which are not of finite type are useful too 4.8 T-invariants are not well defined for general knots Another example of at-invariant which is not of finite type 4.10 Invertible 2-component links which are not invertible through pure knots T-invariants of infinite type for closed braids Gauss diagram invariants of infinite type for pure Legendre fronts without cusps Isotopies with restricted cusp crossing for fronts with exactly two cusps of Legendre knots in ST*JR Bibliography Index

8 Preface Gauss diagram invariants are isotopy invariants of oriented knots in 3- manifolds which are the product of a (not necessarily orientable) surface with an oriented line. The invariants are defined in a combinatorial way using knot diagrams, and they take values in free abelian groups generated by the first homology group of the surface or by the set of free homotopy classes of loops in the surface. There are three main results: 1. The construction of invariants of finite type for arbitrary knots in nonorientable 3-manifolds. These invariants can distinguish homotopic knots with homeomorphic complements. 2. Specific invariants of degree 3 for knots in the solid torus. These invariants cannot be generalized for knots in handlebodies of higher genus, in contrast to invariants coming from the theory of skein modules. 3. We introduce a special class of knots called global knots, in F 2 x lr and we construct new isotopy invariants, called T-invariants, for global knots. Some T-invariants (but not all!) are of finite type but they cannot be extracted from the generalized Kontsevich integral, which is consequently not the universal invariant of finite type for the restricted class of global knots. We prove that T-invariants separate all global knots of a certain type. As a corollary we prove that certain links in 5 3 are not invertible without making any use of the link group!

9 Introduction and announcement This work is an introduction into the world of Gauss diagram invariants. These are isotopy invariants of oriented knots in those 3-manifolds which are the product of a surface with an oriented line. The invariants are defined in a combinatorial way using diagrams of the knot, and they take values in free abelian groups generated by the first homology group of the surface or by the set of free homotopy classes of loops in the surface. Let F 2 be a connected surface, with or without boundary, orientable or not. Let K Y F 2 x ~ be an oriented knot in general position with respect to the (oriented) projection pr : F 2 x ~-+ F 2. In particular, an oriented two-component link in ~ 3 in which one component is the unknot gives rise in a canonical way to a knot in the standard solid torus V = (~ 2 \ (0, 0)) x ~ in ~3. The aim of our work is to construct calculable knot invariants in a systematic way. There are three main results: 1. The construction of invariants of 'finite type' for arbitrary knots in non-orientable 3-manifolds. These invariants take values in Z[H1(F 2 ; Z) EB H 1 (F2; Z)] and they distinguish homotopic knots with homeomorphic complements. (Section 2.9 and 3.4) 2. The construction of specific invariants of degree 3 for knots in the solid torus, which can not be extended to invariants for knots in handlebodies of higher genus. This is in sharp contrast to invariants constructed from the theory of skein modules, which are defined the same way for all handlebodies. (Section 2.6 and 2.8). 3. The construction of new isotopy invariants, called T -invariants, for global knots (Chapter 4). Let F 2 be a compact oriented surface (with or without boundary). A knot type (i.e. a knot up to smooth isotopy) is called a global knot if there is a Marse-Smale vector field v on F 2 and a representative K Y F 2 x ~ of the knot type such that v is transversal to the boundary 8F 2

10 X Gauss Diagram Invariants for Knots and Links each critical point of v has index -1 the projection K '---+ F 2 x IR ~ F 2 is transversal to v. Let G be a quotient group of the group H1(F 2 ; Z)/ < [K] >, where [K] denotes the homology class represented by the oriented knot K. A knot type is called G-pure if it has a representative K such that for each crossing of K '----7 F 2 x IR ~ F 2 each of the two oriented loops obtained by splitting the crossing represents a non trivial element in G. T -invariants are Gauss diagram invariants for G-pure global knots. We define them first as invariants of G-pure knots under isotopy through G pure knots. We prove then that for G-pure global knots K '---+ F 2 x IR our T -invariants are actually invariant under all isotopies. Moreover, they don not depend neither on the chosen vector field v neither on the chosen representative K. Hence, T -invariants are knot invariants in the usual sense. All T -invariants can be calculated with polynomial complexity with respect to the number of crossings of the knot diagrams. However, not all T -invariants are of finite type in the sense of Vassiliev. Moreover, we show that even some T -invariants of finite type cannot be extracted from the generalized K ontsevich integral. We prove that T -invariants separate {in a very effective way) all G-pure global knots for F 2 := T 2 = 8 1 X 8 1 and G := Z/2Z. Let flip : T 2 x IR ~ T 2 x IR be the hyper-elliptic involution on T 2 multiplied by the identity on the lines JR. An oriented knot K '---+ T 2 x ~ is called invertible if it is ambient isotopic to flip (-K) = - flip ( K). Here - K denotes the knot K with reversed orientation. We show that neither the generalized HOMFLY-PT nor the generalized Kauffman polynomial can ever distinguish K from flip (-K). On the other hand we prove the noninvertibility of some global knots in T 2 x IR using T -invariants of finite type. Moreover, knots in T 2 x ~ are in 1-1 correspondence with ordered 3- component links in 8 3 which contain the Hopf link H as a sublink. For example, using a T -invariant of degree 6 for the knot K we show that the link L = KUH ' {see Fig. 0) is not invertible for any chosen orientation

11 Introduction and announcement XI on it. K Fig. 0 L=KUH There are lots of other results and we mention just some of them here: -the construction of an infinite-dimensional space of invariants of degree 2 with values in the group ring Z[H1(F2 ; Z)] (Section 2.4) -isotopy invariants of degree 3 with values in the free abelian group generated by the free homotopy classes of loops in F 2 (Section 2.8) -invariants of degree 3 which detect mutation of two-component links with an unknotted component in JR3 (Section 3.5) -explicit formulas for almost all invariants of degree 3 with Z/3-markings in F 2 x JR, together with a program of Alexander Stoimenow, which calculates them in the case of the solid torus (Sect. 2.6, 2.7.) -invariants of homotopies of isotopies of knots without self-tangencies in a flex (Section 1.8)

12 XII Gauss Diagram Invariants for Knots and Links -the proof that it can be decided whether or not a given knot can be represented by a positive diagram in ffi. 3 (Section 3.8) -flip-invariants for alternating knots in the solid torus (Section 3.11) -the proof that if the fish of Arnold is invertible by a Legendre isotopy with no more than two cusps then both of the two branches, into which the front is divided by the cusps, will go over a cusp (Chapter 5) -construction of invariants of infinite type for pure Legendre fronts which are not invariants of pure isotopy of knots (Section 4.12) All these results are obtained by combinatorial considerations based on the study of Gauss sums associated with Gauss diagrams of knots K Y F 2 x ffi.. We will now describe our method. To a knot K there is naturally associated a diagram on F 2. To each generic diagram corresponds a Gauss diagram, that is, an oriented circle with oriented chords, by connecting points on the circle mapped to a crossing of pr(k) and orienting them from the pre-image of the undercrossing to the pre-image of the overcrossing. (Here we use the orientation of the lines ffi..) Moreover, each chord is marked by a homology class in H1 (F 2 ; Z), which is defined as follows: splitting the diagram with respect to the orientation of the knot in the corresponding crossing p, we obtain two oriented knot diagrams. We associate with the crossing p the homology class represented by those of these diagrams, called K:, which contain the undercross which goes to the overcross (hence the oriented chord together with the corresponding oriented arc in the circle again form an oriented circle). Instead of H1 (F 2 ; Z) we consider sometimes quotients of it or the set of free homotopy classes of loops in F 2. If F 2 is non-orientable then we consider two cases: (wi(f 2 ), [ pr (K)]) = 0 and (wi(f 2 ), [ pr (K)]) # 0. (Here, wi( ) denotes the first Stiefel-Whitney class and (, ) denotes the Kronecker pairing.) In the first case we consider only those crossings p for which (w1 (F 2 ), [ pr (K:)]) = 0. In the second case our formulas mix both types of crossings p. Definition 0.1 A Gauss sum of degree k is a term assigned to a knot diagram, which is of the following form: 2:: function (data, assigned to the crossings) where the sum is taken over all unordered choices of k different crossings in

13 Introduction and announcement XIII the knot diagram, whose arrows in the Gauss diagram form a given subdiagram with given markings. The function is called a weight function. We will denote the summation by the marked subdiagram itself, which we will also call a configuration. A Gauss diagram invariant of degree ::; k is a, perhaps infinite, linear combination of Gauss sums of degree ::; k, such that it is invariant under regular isotopy of the knot (i.e., under Reidemeister moves of type II and III). The combination of Gauss sums of maximal degree in a Gauss sum invariant is called its symbol. A Gauss sum invariant is of degree k if its symbol is a combination of sums of degree k which is either not a Gauss sum invariant (i.e., the symbol is not invariant under regular isotopy) or is already a non-trivial knot invariant (meaning that there are homotopic knots on which it takes different values). Remark concerning notations: Do not confuse our Gauss sums with Gauss sums used in other domains of mathematics. If there is no risk of confusion we also call our Gauss diagram invariants Gauss sum invariants. The expression of a Gauss sum invariant as a combination of Gauss sums is of course not unique. There are lots of Gauss sum identities, which are combinations of Gauss sums that are zero on each knot. The length of a Gauss sum invariant is the minimal number of different configurations in its symbol among all presentations of the invariant as a combination of Gauss sums. Let F 2 be orientable. Then a Gauss sum invariant of degree k is an invariant of finite type in the usual sense of degree ::; k. If F 2 is orientable or if the degree of the Gauss sum invariant is even then it takes values in the group ring Z(H 1(F2 ; z)m) for some m E N otherwise it takes values in z+[h 1(F2 ; z)m). There is only one invariant of degree 1. It is of length 1 and takes values in Z (free homotopy classes of loops in F 2 ), respectively z+ (free homotopy classes of loops along which F 2 is orientable). Let K Y F 2 x JR. be homologous to 0. There are no invariants of degree 2 and oflength 2 with values in Z(H1(F2 ; z)m) form 2: 1. There is exactly one invariant of degree 2 and of length 3 with values in Z(H1(F2 ; Z)2). Surprisingly, there also exists an invariant of degree 2 with values in Z(H1(F2 ; Z)) and which is of infinite length. The invariant of length 3 generalizes to the case of K not homologous to 0 by using the quotient H 1 (F2 ; Z)/ ((K)) instead of Hl(F 2 ; Z). The invariant of infinite length does not generalize! Invariants of degree 3 should live in Z(H1(F2; Z)3). But it becomes too

14 XIV Gauss Diagram Invariants for Knots and Links complicated to find them. Therefore we use instead quotients of H 1(F2;Z) isomorphic to Z/2 or Z/3. Definition 0.2 Let F 2 be orientable and let e = {e 1,...,em} be a basis of HI(F 2 ; Z). To any homology class ~ = L:aiei E H1(F 2 ;Z) we associate the type or In particular, the type [Kh or [Kh of a knot K is defined. Notice that in general the types depend on the chosen basis e, but in the case of an oriented two-component link with an unknotted component in JR3 we have a canonical choice. Instead of a homology class we now attach the type of the homology class to the chord, and we call this a Z/2-marking, respectively Z/3-marking. We say that a configuration has an isolated chord if it has a chord which does not intersect any other chord. For knots in the solid torus we find all invariants of degree 3 with Z/2- markings and which involve only configurations without isolated chords. We find almost all invariants of degree 3 with Z/3- markings. The book is organized as follows: In Section 1 we study the space of diagrams and we construct explicitly cocycles and coboundarys supported in the discriminant consisting of nongeneric diagrams. Knowledge about the discriminant simplifies remarkably the considerations in all other sections of this book. Section 2 contains the main technical part, which consists of large linear systems. Their solutions give the very powerful invariants of degree 3 for knots in F 2 x JR. Section 3 contains the applications and many examples. In Section 4 we develop the concept of G-pure global knots and we construct our T-invariants. These new kind of invariants is the main achievement in this book. Finally, in Section 5 we use our method to study fronts with exactly two cusps of Legendre knots.

15 Introduction and announcement XV Concluding remarks about this book The main question in local knot theory (i.e., knots in 3-space) is whether or not Vassiliev invariants separate knots. The book does not answer this question but it proposes a new direction to the subject. It is shown that there exists a global knot theory quite different from the local one (chapter 4). New invariants for global knots are introduced (some of them are of finite type). They are called T-invariants. These invariants contain (at least some but perhaps all) Vassiliev invariants as a special case. It is shown that T-invariants separate global knots of a certain type! A new question is now whether T-invariants separate all global knots. Gauss diagram formulas for invariants of degree 3 take a lot of place in the book. These formulas have already proven to be useful: they detect mutation of links in JR3 without using cables (chapter 3). No quantum invariant can do this. The tables which are necessary in order to derive the formulas were established by hand, i.e., without the help of a computer. The reason for this is that lots of combinatorial identities were used. We are rather sure that these formulas will find other applications too. The aim of the book is to create its own small world: Gauss diagram invariants for knots in non simply-connected 3-manifolds. There are relations to existing theories like quantum invariants, the Kontsevich integral and finite type invariants, Legendre knot theory. But these relations are not the topic of the book. They are only briefly mentioned in order to show that Gauss diagram invariants add something new for each of these existing theories. Some remarks about history I started these investigations in 1991 with the construction of the invariant of degree 1 published in [F]. In 1993 at the Topology Conference in Siegen (Germany) I gave a talk about the degree 2 invariant and its inequality for positive closed braids. At this time I called the invariants small state sums, influenced by Kauffman's state model of the Jones polynomial. In 1996 Alexander Stoimenow wrote his program which made it possible to calculate, for sophisticated examples, the invariants of degree 3 for knots in the solid torus. In 1998 I found a possibility of generalizing the approach in order to obtain new invariants of infinite type for a naturally restricted sort of isotopies (pure knots and pure isotopies). Finally, in spring 2000 I have realized that invariants under pure isotopies

16 XVI Gauss Diagram Invariants for Knots and Links are in fact isotopy invariants in the case of global knots. The result is a new sort of invariants, called T-invariants. In the meantime, of course, other people had similar ideas. I should mention the works of Arnold about immersed planar curves [A], Lannes about formulas for the Vassiliev invariants v2 and v3 [La], Goryunov about invariants of finite type in the solid torus [Go], and, especially, the work of Polyak and Viro about the formalism of Gauss diagram invariants [PV]. I profited from all these works, and, of course, from the pioneering work of Vassiliev [V],[V2] in a conceptual sense. More recent work of Andersen, Mattes, Reshetikhin [AMR] and Goussarov, Polyak, Viro [PVG] is also related to our work. Finally, I should indicate that I adapted the fine terminology of Polyak, Viro and of Stoimenow, especially the notions of Gauss diagrams, Gauss sums and Gauss diagram invariants. Acknowledgments. I am very grateful to Alexander Stoimenow. His program for calculating the invariants is a powerful tool. In addition to this, we had lots of interesting discussions. He and Klaus Mahnke also helped me in resolving the large linear systems. I wish to thank Michel Boileau, Joan Birman and my wife Severine for believing in my work over a long time. I am grateful to J0rgen Andersen and to Michael Polyak for interesting discussions. Finally, I wish to thank Elena Alferova for her patience in typing this manuscript. Thomas Fiedler Toulouse, May 2001