Contributions in Mathematical and Computational Sciences Volume 1
Editors Hans Georg Bock Willi Jäger
For other titles published in this series, go to www.springer.com/series/8861
[email protected] Markus Banagl Denis Vogel Editors
The Mathematics of Knots
Theory and Application
[email protected] Editors Markus Banagl Denis Vogel Dept. of Mathematics Dept. of Mathematics Heidelberg University Heidelberg University Im Neuenheimer Feld 288 Im Neuenheimer Feld 288 69120 Heidelberg 69120 Heidelberg Germany Germany [email protected] [email protected]
ISBN 978-3-642-15636-6 e-ISBN 978-3-642-15637-3 DOI 10.1007/978-3-642-15637-3 Springer Heidelberg Dordrecht London New York
Mathematics Subject Classification (2010): 57M25, 57Q45, 92C37, 81T45
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[email protected] Preface to the Series
Contributions in Mathematical and Computational Sciences
Mathematical theories and methods and effective computational algorithms are cru- cial in coping with the challenges arising in the sciences and in many areas of their application. New concepts and approaches are necessary in order to overcome the complexity barriers particularly created by nonlinearity, high-dimensionality, mul- tiple scales and uncertainty. Combining advanced mathematical and computational methods and computer technology is an essential key to achieving progress, often even in purely theoretical research. The term mathematical sciences refers to mathematics and its genuine sub-fields, as well as to scientific disciplines that are based on mathematical concepts and meth- ods, including sub-fields of the natural and life sciences, the engineering and so- cial sciences and recently also of the humanities. It is a major aim of this series to integrate the different sub-fields within mathematics and the computational sci- ences, and to build bridges to all academic disciplines, to industry and other fields of society, where mathematical and computational methods are necessary tools for progress. Fundamental and application-oriented research will be covered in proper balance. The series will further offer contributions on areas at the frontier of research, providing both detailed information on topical research, as well as surveys of the state-of-the-art in a manner not usually possible in standard journal publications. Its volumes are intended to cover themes involving more than just a single “spectral line” of the rich spectrum of mathematical and computational research. The Mathematics Center Heidelberg (MATCH) and the Interdisciplinary Center for Scientific Computing (IWR) with its Heidelberg Graduate School of Mathemat- ical and Computational Methods for the Sciences (HGS) are in charge of providing and preparing the material for publication. A substantial part of the material will be acquired in workshops and symposia organized by these institutions in topical areas of research. The resulting volumes should be more than just proceedings collect- ing papers submitted in advance. The exchange of information and the discussions during the meetings should also have a substantial influence on the contributions.
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[email protected] vi Preface to the Series
Starting this series is a venture posing challenges to all partners involved. A unique style attracting a larger audience beyond the group of experts in the subject areas of specific volumes will have to be developed. The first volume covers the mathematics of knots in theory and application, a field that appears excellently suited for the start of the series. Furthermore, due to the role that famous mathematicians in Heidelberg like Herbert Seifert (1907Ð1996) played in the development of topology in general and knot theory in particular, Hei- delberg seemed a fitting place to host the special activities underlying this volume. Springer Verlag deserves our special appreciation for its most efficient support in structuring and initiating this series.
Heidelberg University, Germany Willi Jäger Hans Georg Bock
[email protected] Preface
This volume is based on the themes of, and records advances achieved as a re- sult of, the Heidelberg Knot Theory Semester, held in winter 2008/09 at Heidelberg University under the sponsorship of the Mathematics Center Heidelberg (MATCH), organized by M. Banagl and D. Vogel. In the preceding summer semester an intro- ductory seminar on knots aimed at providing non-experts and young mathematicians with some of the foundational knowledge required to participate in the events of the winter semester. These comprised expository lecture series by several leading ex- perts, representing rather diverse aspects of knot theory and its applications, and a concluding workshop held December 15 to 19, 2008. Knots seem to be a deep structure, whose peculiar feature it is to surface unex- pectedly in many different and a priori unrelated areas of mathematics and the nat- ural sciences, such as algebra and number theory, topology and geometry, analysis, mathematical physics (in particular statistical mechanics), and molecular biology. Its relevance in topology, apart from its intrinsic interest, is partly due to the fact that every closed, oriented 3-manifold can be obtained by surgery on a framed link in the 3-sphere. Modern topology has also obtained information on high-dimensional knots, that is, embeddings of an n-sphere in an (n+2)-sphere with n larger than one. In algebra, representations of quantum groups lead to a multitude of knot invariants. Based on ideas of B. Mazur in number theory, one can assign to two prime ideals of a number field a linking number in analogy with classical knot theory. This number- theoretic linking number plays a role in studying the structure of Galois groups of certain extensions of the number field. Analysis touches on knot theory by means of operator algebras and their connection to the Jones polynomial. As far as geometry is concerned, results by Fenchel on the curvature of a closed space curve date back to the 1920s. Milnor showed in 1949 that the curvature must exceed 4π if the curve is knotted. One also considers “real” knots as physical objects in 3-space and stud- ies various natural energy functionals on them. Sums taken over all states of suitable models originating in statistical mechanics, describing large ensembles of particles, can express knot invariants such as L. Kauffman’s bracket polynomial. The dis- covery of the Jones polynomial entailed ties with mathematical physics based on a curious congruity of five relations, namely the Artin-relation in braid groups, a fun- damental relation in certain operator algebras due to Hecke, the third Reidemeister
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[email protected] viii Preface move, the classical Yang-Baxter equation, and its quantum version. This lead to the construction of topological quantum field theories by Witten and Atiyah. Cellular DNA is a long molecule, which may be closed (as e.g. the genome of certain bacte- ria) and knotted or linked with other DNA strands. Enzymes such as topoisomerase or recombinase operate on DNA changing the topological knot or link type. The objective of the Heidelberg Knot Theory Semester was to do justice to this diversity by bringing together representatives of most of the above research avenues, accompanied by the hope that such a meeting might foster inspiration and synergy across the various questions and approaches. Certainly, a fairly comprehensive por- trait of the current state-of-the-art in knot theory and its applications emerged as a result. Four lecture series were given: DeWitt Sumners gave 5 lectures on scientific applications of knot theory, discussing DNA topology, a tangle model for DNA site-specific recombination, random knotting, topoisomerase, spiral waves and vi- ral DNA packing. Kent Orr’s 3 lectures explained knot concordance and surgery techniques, while Louis Kauffman’s 2 lectures introduced virtual knots and detailed parallels to elementary particles. The topic of Masanori Morishita’s 6 lectures were the aforementioned analogies between knot theory and number theory. The 21 speakers of the final workshop “The Mathematics of Knots” reported on a variety of interesting current developments. Many of these accounts are mirrored in the papers of the present volume. Among the low-dimensional topics were virtual knots and associated invariants such as arrow and Jones polynomials, the HOMFLY polynomial, questions about Dehn filling, Legendrian knots, Khovanov homology, surface knots, slice knots, fibered knots and property R, colorings by metabelian groups, singular knots, Gram determinants of planar curves, as well as geometric structures such as surfaces associated with knots, and the fibering of 3-manifolds when the product of the manifold with a circle is known to be symplectic. High- dimensional topics concerned the Cohn noncommutative localization of rings and its application to knots via algebraic K- and L-theory, as well as high-dimensional non- locally flat embeddings and the role of knot theory vis-à-vis transformation groups. Scientific talks discussed random knotting, viral DNA packing, and the topology of DNA-protein interactions. We wish to extend our sincere thanks to the contributors of this volume and to all participants of the Heidelberg Knot Theory Semester, especially to the lecturers giving mini-courses, for the energy and time they have devoted to this event and the preparation of the present collection. Paul Seyfert receives the editors’ thanks for technical help in typesetting this volume. Furthermore, we are grateful to Dorothea Heukäufer for her efficient handling of numerous logistical issues. Finally, we would like to express our gratitude to Willi Jäger and MATCH, whose financial support made the Heidelberg Knot Theory Semester possible.
Heidelberg University, Germany Markus Banagl Denis Vogel
[email protected] Contents
1 Knots, Singular Embeddings, and Monodromy ...... 1 Markus Banagl, Sylvain E. Cappell, and Julius L. Shaneson
2 Lower Bounds on Virtual Crossing Number and Minimal Surface Genus ...... 31 Kumud Bhandari, H.A. Dye, and Louis H. Kauffman
3 A Survey of Twisted Alexander Polynomials ...... 45 Stefan Friedl and Stefano Vidussi
4 On Two Categorifications of the Arrow Polynomial for Virtual Knots ...... 95 Heather Ann Dye, Louis Hirsch Kauffman, and Vassily Olegovich Manturov
5 An Adelic Extension of the Jones Polynomial ...... 125 Jesús Juyumaya and Sofia Lambropoulou
6 Legendrian Grid Number One Knots and Augmentations of Their Differential Algebras ...... 143 Joan E. Licata
7 Embeddings of Four-valent Framed Graphs into 2-surfaces .....169 Vassily Olegovich Manturov
8 Geometric Topology and Field Theory on 3-Manifolds ...... 199 Kishore Marathe
9 From Goeritz Matrices to Quasi-alternating Links ...... 257 Józef H. Przytycki
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[email protected] x Contents
10 An Overview of Property 2R ...... 317 Martin Scharlemann
11 DNA, Knots and Tangles ...... 327 De Witt Sumners
Workshop Talks ...... 355
[email protected] Chapter 1 Knots, Singular Embeddings, and Monodromy
Markus Banagl, Sylvain E. Cappell, and Julius L. Shaneson
Abstract The Goresky-MacPherson L-class of a PL pseudomanifold piecewise- linearly embedded in a PL manifold in a possibly nonlocally flat way, can be computed in terms of the Hirzebruch-Thom L-class of the manifold and twisted L-classes associated to the singularities of the embedding, as was shown by Cappell and Shaneson. These formulae are refined here by analyzing the twisted classes. We treat the case of Blanchfield local systems that extend into the singularities as well as cases where they do not extend. In the latter situation, we consider fibered embeddings of strata and 4-dimensional singular sets, using work of Banagl. Rho- invariants enter the picture.
1.1 Introduction
Let Mn+2 be a closed, oriented, connected PL manifold of dimension n+2 and Xn a closed, oriented, connected PL pseudomanifold of dimension n.Leti : X→ M be a
The first author was partially supported by a grant of the Deutsche Forschungsgemeinschaft. The second and third authors were partially supported by grants of the Defense Advanced Research Projects Agency. M. Banagl () Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany e-mail: [email protected]
S.E. Cappell Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA e-mail: [email protected]
J.L. Shaneson Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA e-mail: [email protected]
M. Banagl, D. Vogel (eds.), The Mathematics of Knots, 1 Contributions in Mathematical and Computational Sciences 1, DOI 10.1007/978-3-642-15637-3_1, © Springer-Verlag Berlin Heidelberg 2011 [email protected] 2 M. Banagl et al.
∗ 1 2 not necessarily locally flat PL embedding. Let L = 1 + L (p1) + L (p1,p2) +··· be the total Hirzebruch L-polynomial,
1 1 L1(p ) = p ,L2(p ,p ) = (7p − p2),.... 1 3 1 1 2 45 2 1 Let P(M)∈ H ∗(M; Z) be the total Pontrjagin class of M and the Euler class χ ∈ 2 H (M; Z) be the Poincaré dual of i∗[X]∈Hn(M; Z), where [X] is the fundamental class of X. Set
∗ ∗ 2 −1 L∗(M, X) =[X]∩i L (P (M) ∪ (1 + χ ) ) ∈ H∗(X; Q).
Recall that the sequence L1,L2,... of polynomials is the multiplicative sequence associated to the even power series defined by x/tanh(x). Thus
∗ χ 1 1 L (1 + χ2) = = 1 + χ2 − χ4 ±··· tanh(χ) 3 45 and by the multiplicativity of {Lj },
∗ − tanh(χ) 1 2 L ((1 + χ2) 1) = = 1 − χ2 + χ4 ∓··· . χ 3 15
Hence the above defining expression for L∗(M, X) may alternatively be written as ∗ tanh i χ ∗ ∗ L∗(M, X) =[X]∩ ∪ i L (P M) i∗χ 1 ∗ 2 ∗ ∗ ∗ =[X]∩ 1 − i χ2 + i χ4 ∓··· ∪ i L (P M) . 3 15
When this formula is pushed on into M, one obtains ∗ tanh χ ∗ i∗L∗(M, X) = i∗ [X]∩i ∪ L (P M) χ tanh χ ∗ = i∗[X]∩ ∪ L (P M) χ tanh χ = ([M]∩χ)∩ ∪ L∗(P M) χ ∗ =[M]∩(tanh χ ∪ L (P M)).
If the embedding is nonsingular, that is, X is a locally flat submanifold, then
L∗(X) = L∗(M, X),
[email protected] 1 Knots, Singular Embeddings, and Monodromy 3 where L∗(X) is the Poincaré dual of the Hirzebruch L-class of X. In particular, the signature σ(X)= L0(X) is given by
σ(X)= L0(M, X). If the embedding is singular, the singularities of X and the singularities of the embedding induce a stratification of the pair (M, X). Under the assumption that there are no strata of odd codimension, it was shown in [CS91] that the Goresky- MacPherson L-class L∗(X) ∈ H∗(X; Q) of X, defined using middle-perversity in- tersection homology, can be computed as = − ; BR L∗(X) L∗(M, X) iV ∗L∗(V V ), (1.1) V ∈X where the sum ranges over all connected components V of pure strata of X that have codimension at least two, iV : V→ X is the inclusion of the clo- ; BR ∈ ; Q sure V of V into X and L∗(V V ) H∗(V ) is the Goresky-MacPherson L- BR class of V twisted by a local coefficient system V . This local system is en- BR ⊗ BR → R dowed with a nonsingular symmetric or skew-symmetric form V V and arises as Trotter’s “scalar product” [Tro73] of a certain Blanchfield local sys- B ⊗ Bop → Q = Q[ −1] tem V V (t)/, t,t . The systems are defined on V and do not in general extend as local systems to the closure V . They do, of course, ex- tend as intersection chain sheaves by applying Deligne’s pushforward/truncation- BR ; BR formula to V , and L∗(V V ) is defined as the L-class of this self-dual sheaf complex on V . (For an introduction to the L-class of self-dual sheaves see [Ban07].) In the present paper, we refine formula (1.1) by computing the twisted classes ; BR BR L∗(V V ) further. Two cases are to be distinguished: The systems V either extend as local systems from V to V or they do not. In the former situation, the results of [BCS03] apply and yield the formula (Theorem 6) = − [BR] ∩ L∗(X) L∗(M, X) iV ∗(ch V K L∗(V )), (1.2) V ∈X where the modification ch of the Chern character is given by precomposing with the = ◦ 2 [BR] second Adams operation, ch ch ψ and V K denotes the K-theory signature BR BR of V , an element of KO(X) if the form on V is symmetric, and of KU(X) if it is skew-symmetric. In the situation of nonextendable systems, formulae of type (1.2), even when the right hand side is defined, cease to hold as counterexamples of [Ban08] show. The main results presented here, then, are concerned with un- ; BR BR derstanding the twisted signatures σ(V V ) when V does not extend as a local system into the singularities of V . Theorem 10 asserts that
σ(X)= L0(M, X) when all embeddings V − V→ V are locally flat spherical fibered knots. In par- n+2 n+2 ticular if M = S is a sphere, we have σ(X)= 0, since L0(S ,X)= 0. The
[email protected] 4 M. Banagl et al. remaining results all assume that i : X→ M has a 4-dimensional singular set such that the V are 4-manifolds and the bottom stratum consists of locally flat 2-spheres (see Examples 1 and 2). If the 2-spheres have zero self-intersection numbers and BR = =− V is positive (V 1) or negative (V 1) definite of rank rV , then σ(X)= L0(M, X) − V rV σ(V), V ⊂X4−X2 with V ranging over all connected components V of the pure 4-stratum (Theorem 7). Again we obtain a corollary for the case where M is a sphere: σ(X)+ V rV σ(V)= 0. V ⊂X4−X2 Similar corollaries for embeddings in spheres can be deduced for the following re- sults as well. More generally, if the structure group of the form on V is O(pV ,qV ), then = − − − 2 − BC [ ] σ(X) L0(M, X) (pV qV )σ (V) 2(c1 2c2)( V ), V , V ⊂X4−X2 V ⊂X4−X2 2 − 4 ; Z where 2(c1 2c2) is an H (V )-valued characteristic class (Theorem 8). As a corollary (Corollary 4) we deduce that σ(X)− L0(M, X) is divisible by 8 if every V is a 4-sphere. When the 2-spheres have nonzero self-intersection numbers, then rho-invariants enter. Theorem 9 for positive, say, definite forms asserts that
σ(X)= L (M, X) 0 nV − + BC| [ 2]2 − rV σ(V) (c-rk( V Li ) sign Si ραi (pi,qi)) , V ⊂X4−X2 i=1 where σ(V)denotes the (Novikov-) signature of the exterior of the 2-spheres