Selected Title s i n Thi s Serie s

24 Jan e Gilman , Willia m W . Menasco , an d Xiao-Son g Lin , Editors , Knots , Braids , and Mappin g Clas s Groups—Paper s Dedicate d t o Joa n S . Birman, 200 1 23 Cumru n Vaf a an d S.-T . Yau , Editors , Winte r Schoo l o n Mirro r Symmetry , Vecto r Bundles an d Lagrangia n Submanifolds , 200 1 22 Carlo s Berenstein , Der-Che n Chang , an d Jingzh i Tie , Laguerr e Calculu s an d It s Applications o n the Heisenber g Group , 200 1 21 Jiirge n Jost , Bosoni c Strings : A Mathematical Treatment , 200 1 20 L o Yan g an d S.-T . Yau , Editors , Firs t Internationa l Congres s o f Chines e Mathematicians, 200 1 19 So-Chi n Che n an d Mei-Ch i Shaw , Partia l Differentia l Equation s i n Severa l Comple x Variables, 200 1 18 Fangyan g Zheng , Comple x Differentia l Geometry , 200 0 17 Le i Gu o an d Stephe n S.-T . Yau , Editors , Lecture s o n Systems , Control , an d Information, 200 0 16 Rud i Weikar d an d Gilber t Weinstein , Editors , Differentia l Equation s an d Mathematical Physics , 200 0 15 Lin g Hsia o an d Zhoupin g Xin , Editors , Som e Curren t Topic s o n Nonlinea r Conservation Laws , 200 0 14 Jun-ich i Igusa , A n Introductio n t o the Theor y o f Loca l Zet a Functions , 200 0 13 Vasilio s Alexiade s an d Georg e Siopsis , Editors , Trend s i n Mathematica l Physics , 1999 12 Shen g Gong , Th e Bieberbac h Conjecture , 199 9 11 Shinich i Mochizuki , Foundation s o f p-adic Teichmulle r Theory , 199 9 10 Duon g H . Phong , Lu c Vinet , an d Shing-Tun g Yau , Editors , Mirro r Symmetr y III , 1999 9 Shing-Tun g Yau , Editor , Mirro r Symmetr y I , 199 8 8 Jiirge n Jost , Wilfri d Kendall , Umbert o Mosco , Michae l Rockner , and Karl-Theodo r Sturm , Ne w Direction s i n Dirichle t Forms , 199 8 7 D . A . Buel l an d J . T . Teitelbaum , Editors , Computationa l Perspective s o n Numbe r Theory, 199 8 6 Harol d Levine , Partia l Differentia l Equations , 199 7 5 Qi-ken g Lu , Stephe n S.-T . Yau , an d Anatol y Libgober , Editors , Singularitie s an d Complex Geometry , 199 7 4 Vyjayanth i Char i an d Iva n B . Penkov , Editors , Modula r Interfaces : Modula r Li e Algebras, Quantu m Groups , an d Li e Superalgebras, 199 7 3 Xia-X i Din g an d Tai-Pin g Liu , Editors , Nonlinea r Evolutionar y Partia l Differentia l Equations, 199 7 2.2 Willia m H . Kazez , Editor , Geometri c Topology , 199 7 2.1 Willia m H . Kazez , Editor , Geometri c Topology , 199 7 1 B . Green e an d S.-T . Yau , Editors , Mirro r Symmetr y II , 199 7 This page intentionally left blank Knots, Braids, and Mappin g Class Groups—Papers Dedicate d to Joan S . Birman This page intentionally left blank https://doi.org/10.1090/amsip/024 AMS/IP Studies in Advanced Mathematics

Volume 24

Knots, Braids, and Mappin g Class Groups—Papers Dedicate d to Joan S . Birman Proceedings o f a Conferenc e in Low Dimensional Topolog y in Honor o f Joan S . Birman's 70t h Birthda y March 14-15 , 199 8 Columbia Universit y New York, New York Jane Gilman, William W. Menasco, and Xiao-Song Lin, Editors

American Mathematical Society • International Press Shing-Tung Yau , Genera l Edito r

2000 Mathematics Subject Classification. Primar y 42C40 .

Library o f Congres s Cataloging-in-Publicatio n Dat a Knots, braids , an d mappin g clas s groups—paper s dedicate d t o Joa n S . Birma n : proceeding s o f a conferenc e o n lo w dimensiona l topolog y i n hono r o f Joa n S . Birman' s 70t h birthday , Marc h 14-15, 1998 , Columbi a University , Ne w York , Ne w Yor k / Jan e Gilman , Willia m W . Menasco , and Xiao-Son g Lin , editors . p. cm . — (AMS/I P studie s i n advance d mathematics , ISS N 1089-328 8 ; v. 24 ) Includes bibliographica l references . ISBN 0-8218-2966- 1 (alk . paper ) 1. Lo w dimensiona l topology—Congresses . I . Birman , Joa n S. , 1927 - II . Gilman , Jane , 1945- III . Menasco , Willia m W. , 1954 - IV . Lin , Xiao-Song , 1957 - V . Series .

QA612.14.K58 200 1 514'.22—dc21 2001053661

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© 200 1 b y the America n Mathematica l Societ y an d Internationa l Press . Al l rights reserved . The America n Mathematica l Societ y an d Internationa l Pres s retai n al l right s except thos e grante d t o th e Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t URL : http://www.ams.org / Visit th e Internationa l Pres s hom e pag e a t URL : http://www.intlpress.com / 10 9 8 7 6 5 4 3 2 1 0 6 0 5 0 4 0 3 0 2 0 1 Table o f Contents

Preface i x Joan S . Birman: Publication s an d Ph.D . These s Supervise d xii i Photos xvi i J. Cantarella , D . DeTurck, an d H . Gluck , Upper Bounds for the Writhing of Knots and the Helicity of Vector Fields 1 O.T. Dasbac h an d B.S . Mangum, The Automorphism Group of a Free Group is not Subgroup Separable 2 3 R. Ghrist , Configuration Spaces and Braid Groups on Graphs in Robotics . 29 J. Gilman , Alternate Discreteness Tests 4 1 S.P. Humphries , Intersection-number Operators for Curves on Discs and Chebyshev Polynomials 4 9 O. Kharlampovich an d A . Myasnikov , Implicit Function Theorem over Free Groups and Genus Problem 7 7 M.E. Kidwel l an d T.B . Stanford , On the z-Degree of the Kauffman Polynomial of a Tangle Decomposition 8 5 W. Li , Knot Invariants from Counting Periodic Points 9 5 X.-S. Li n an d Z . Wang, Random Walk on Knot Diagrams, Colored Jones Polynomial and Ihara-Selberg Zeta Function 10 7 F. Luo , Some Applications of a Multiplicative Structure on Simple Loops in Surfaces 12 3 W.W. Menasco , Closed Braids and Heegaard Splittings 13 1 J.H. Przytycki , Homotopy and q-Homotopy Skein Modules of 3-Manifolds: An Example in Algebra Situs 14 3 T.B. Stanfor d an d R . Trapp , On Knot Invariants which are not of Finite Type 17 1

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On Marc h 1 4 & 15 , 1998 , a conferenc e i n low-dimensiona l topolog y i n hono r of Professor Joa n Birman' s 70t h birthday wa s held i n Columbi a Universit y i n Ne w York City . Th e organizin g committe e o f tha t conferenc e consiste d o f Vladimi r Arnold, Dav e Bayer, Elizabeth Finkelstein , Rober t Friedma n (chair) , Jane Gilman , Vaughan Jones , Xiao-Son g Lin , Bria n Mangum , an d Willia m Menasco . A gran t from NSF gave partial support fo r the conference. Th e warm hospitality o f Barnard College an d th e Mathematic s Departmen t o f Columbi a Universit y a s wel l a s th e talks given by Colin Adams, Robert Ghrist, Vaughan Jones, Xiao-Song Lin, William Menasco, Joze f Przytyski , Carolin e Series , and Arkad y Vaintro b mad e this confer - ence a memorable even t fo r it s on e hundred plu s participants .

There ar e a number o f specialties i n low-dimensional topolog y that ca n find i n their "famil y tree " a commo n ancestr y i n th e theor y o f surfac e mappings . Thes e include kno t theor y a s studie d throug h th e us e o f brai d representation s an d 3 - manifolds a s studie d throug h th e us e o f Heegaar d splittings . Th e stud y o f th e surface mappin g clas s grou p (th e modula r group ) i s o f course a ric h subjec t i n it s own right , wit h relation s t o man y differen t fields o f mathematic s an d theoretica l physics. Bu t it s mos t direc t an d remarkabl e manifestatio n i s probably i n the vas t area o f low-dimensional topology . Althoug h the scene of this area has been change d dramatically an d experience d significant expansio n sinc e the original publication o f her semina l wor k "Braids , Links , an d Mappin g Clas s Groups " (1974) , Professo r Joan Birman ha s played a prominent rol e during this entire period. Th e conferenc e in honor o f Professor Joa n Birman brought together mathematicians whose research span man y specialtie s al l o f such a common lineage . Th e papers i n this volume ar e contributed b y th e participant s o f thi s Birmanfest . Th e topic s covere d ar e quit e diverse. Bu t w e believe that the y reflec t wel l the ai m an d spiri t o f the conference , that i s to explor e ho w thes e variou s specialtie s i n low-dimensiona l topolog y hav e diverged i n the past 20-2 5 years as well as to explore common threads and potentia l future direction s o f development . Wit h dee p admiratio n an d appreciatio n o f he r contribution to low-dimensional topology, we dedicate this volume to Professor Joa n Birman.

For the convenienc e o f the reader, w e introduce briefl y th e topic o f each pape r in this volume . The pape r "Uppe r bound s fo r th e writhin g o f knots an d th e helicit y o f vecto r fields" b y J . Cantarella , D . DeTurck , an d H . Gluc k lie s i n th e field o f "geometri c knot theory" , wher e the mai n attentio n i s focused o n the possibl e geometri c shap e of a knot i n the 3-space . Th e writhing o f a geometric kno t i s obtained b y applyin g the Gaus s linkin g integra l t o a singl e curve , an d i t i s relate d t o th e helicit y o f vector fields i n 3-spac e i n th e wor k o f V . Arnold . I n thi s paper , a n uppe r boun d for th e writhin g o f a kno t b y th e 4/ 3 powe r o f th e quotien t o f it s lengt h an d it s thickness i s obtained. Simila r upper bounds fo r the helicity also holds. Othe r shap e upper bound s fo r helicit y an d optima l vecto r fields realizin g thes e uppe r bound s are discusse d a s well . The paper "Th e automorphism group of a free group is not subgroup separable " by O . Dasbac h an d B . Mangum exhibit s a special grou p theoreti c propert y whic h distinguishes the brai d grou p B^ wit h B2 an d B%. The property i s called subgrou p ix X separable, whic h i s to us e finite inde x subgroup s t o separat e a finitel y generate d subgroup an d a n elemen t no t i n thi s subgroup . I t i s know n tha t B2 an d B 3 ar e subgroup separable , bu t i t i s shown i n this pape r tha t B^ i s not so . The pape r "Configuratio n space s an d brai d group s o n graph s i n robotics " b y R. Ghris t originate s th e stud y o f brai d group s o n graphs . Som e basi c result s ar e obtained, lik e the vanishin g o f higher homotop y group s o f the configuratio n space s in question , bound s o n homolog y dimensio n o f th e configuratio n spaces , an d th e torsion-free-ness o f the brai d groups . Backgroun d i n designing saf e contro l schem e for robot s i s discussed . The pape r "Alternat e discretenes s tests " b y J . Gilma n i s about a faste r algo - rithm fo r decidin g whethe r tw o element s o f P5L(2,7£), th e isometr y grou p o f th e hyperbolic plane , generat e a non-elementar y discret e subgroup . Som e ne w shor t tests fo r discretenes s ar e obtained a s a corollary . The pape r "Intersection-numbe r operator s fo r curve s o n disc s an d Chebyshe v polynomials" b y S . Humphries studie s th e algebrai c an d combinatoria l propertie s of the intersection s o f essential simpl e close d curve s o n a punctured disk . The pape r "Implici t functio n theore m ove r fre e group s an d genu s problem " by O . Kharlampovic h an d A . Myasniko v give s som e application s o f th e implici t function theore m fo r fre e groups . Th e ide a o f a n invers e functio n theore m fo r free group s arise s i n Birman' s earl y work . Th e pape r i n thi s volum e contain s a n application o f concepts developed b y Kharlampovich an d Myasniko v i n their recen t solution t o the Tarsk i conjecture . Th e genu s proble m ask s fo r som e contro l o n th e solution o f equations ove r a fre e group . I t i s shown here using the implici t functio n theorem that suc h contro l ma y no t b e possibl e i n som e cases . The paper "O n the z-degre e o f the Kauffma n polynomia l o f a tangle decompo - sition" b y M . Kidwel l an d T . Stanfor d analyze s th e relatio n betwee n th e ^-degre e of Kauffma n polynomia l an d th e length s o f arc s i n a kno t digra m (th e length s o f arcs in an alternating kno t digra m ar e all one). Th e analysis i s carried out b y usin g tangle decomposition s o f knot diagrams . The pape r "Kno t invariant s fro m countin g periodi c points " b y W . L i shows , among som e othe r result s o f the sam e nature , tha t th e generatin g functio n o f "it - erated kno t signatures " ha s a positiv e radiu s o f convergence . Th e classica l kno t signature ca n b e interprete d a s th e Lefschetz numbe r o f a brai d diffeomorphism . The Lefschet z numbe r o f the iteratio n o f the brai d diffeomorphis m i s shown to b e a kno t invariant . The pape r "Rando m wal k o n kno t diagrams , colore d Jone s polynomia l an d Ihara-Selberg zet a function " b y X.-S . Li n an d Z . Wan g use s a mode l o f rando m walk o n knot diagram s to study kno t polynomials . I n this context, th e relationshi p between th e Alexande r polynomia l an d th e colore d Jone s polynomia l addresse d i n the so-calle d Melvin-Morto n conjectur e become s transparent . The pape r "Som e application s o f a multiplicative structur e o n simpl e loop s i n surfaces" b y F . Lu o use s th e multiplicativ e structur e o n simpl e loop s o n surface s to sho w tha t th e intersectio n numbe r o f simpl e loop s o n surface s ha s a certai n convexity. The paper "Close d braids and Heegaard splittings" b y W. Menasco investigate s a strateg y fo r constructin g 3-manifold s havin g non-equivalen t strongl y irreducibl e Heegaard splittings . Th e strateg y combine s technique s involvin g frame d link s an d a calculu s o n close d braid s develope d b y Birman an d Menasco . xi

The pape r "Homotop y an d q-homotop y skei n module s o f 3-manifolds : a n ex - ample i n Algebr a Situs " b y J . Przytyck i i s a survey o f the branc h o f mathematic s called Algebra Situ s promoted b y the author. I t concentrates o n a relatively simpl e example o f th e g-homotop y skei n module . Despit e o f it s simpl e appearance , thi s skein modul e ha s alread y connecte d wit h man y othe r mathematica l ingredients . The pape r "O n kno t invariant s whic h ar e no t o f finit e type " b y T . Stanfor d and R . Trap p give s som e criteri a fo r a kno t invarian t no t bein g o f finite type . A generalization o f the notio n o f finite type invariant s i s also introduced .

We would lik e to take this opportunity t o thank th e authors o f this volum e fo r their contribution s an d patience . W e believ e tha t w e al l shar e th e commo n wis h that Professo r Joa n Birma n wil l continu e he r productiv e caree r a s par t o f a lon g and health y life .

Jane Gilma n William W. Menasc o Xiao-Song Li n December 15 , 199 9 This page intentionally left blank Joan S . Birman: Publication s an d Ph.D. These s Supervise d

Publications: 1. "O n braid groups" , Com. Pure and App. Math., 22(1969) , pp. 41-72 . 2. "Mappin g clas s groups an d thei r relationshi p t o brai d groups" , Com. Pure and App. Math., 2 2 (1969) , pp. 212-238 . 3. "Automorphism s o f the fundamental grou p of a closed, orientable 2-manifold" , Proc. AMS, 21(1969) , pp. 351-354 . 4. "Abelia n quotients o f the mapping class group of a 2-manifold", Bull. AMS, 76(1970), pp. 147-150 . 5. "O n Siegal' s modular group" , Math. Annalen, 19 1 (1971) , pp.59-68. 6. "Mappin g clas s group s o f close d surface s a s coverin g spaces" , (wit h H.M . Hilden), Annals of Math Studies, 6 6 (1971) , 81-115. 7. "O n th e homeotop y grou p o f a non-orientabl e surface" , (wit h D . Chilling - worth), Proc. Cambridge Phil. Soc., 71(1972) , pp. 437-448 . 8. " A norma l for m i n th e homeotop y grou p o f a surfac e o f genu s 2 , wit h a n application t o 3-manifolds" , Proc. AMS, 34(1972) , pp. 397-384 . 9. "O n lifting an d projecting homeomorphisms" , (wit h H.M. Hilden), Archives of Math., 23(1972) , pp. 428-434 . 10. "Isotopie s o f homeomorphism s o f Rieman n surface s an d a theore m abou t Artin's brai d group" , (wit h H.M . Hilden) , Bull AMS, 78(1972) , pp. 1002 - 1004. 11. "Isotopie s o f homeomorphism s o f Rieman n surfaces" , (wit h H.M . Hilden) , Annals of Math., 97(1973) , pp. 424-439 . 12. "Poincare' s conjectur e an d th e homeotop y grou p o f a closed , orientabl e 2 - manifold", J. Australian Math. Soc, XVII(1974) , pp . 214-221 . 13. "A n invers e functio n theore m fo r fre e groups" , Proc. AMS, 41(1983) , pp . 634-638. 14. "Th e homeomorphis m proble m fo r 5 3", (wit h H.M . Hilden) , Bull. AMS, 79(1973), pp. 1006-1010 . 15. "Pla t presentatio n fo r lin k groups", Com. Pure and Applied Math., XXV I (1973), pp. 673-678 . 16. Braids , link s an d mappin g clas s groups, Ann. Math . Studies , vol. 82 , Prince- ton Universit y Press , 1974 . 17. "Heegaar d splitting s o f branche d coverin g o f 5 3", (wit h Hug h M . Hilden) , Trans. AMS, 21 3 (1975) , pp. 315-353 . 18. "O n the equivalence of Heegaard splittings of closed, orientable 3-manifolds" , Annals of Math Studies, vol . 8 4 (ed. L.P . Neuwirth), Princeton Univ . Press , 1975, pp.137-164 . xiii xiv JOA N S . BIRMAN : PUBLICATION S AN D PH.D . THESE S SUPERVISE D

19. " A note o n the representatio n o f simply-connected 3-manifold s a s branche d covering space s o f S3", Proc. AMS, 55(1976) , pp. 440-442 . 20. "O n th e stabl e equivalenc e o f Heegaar d splitting s o f pla t presentatio n o f links", Canad. J. Math., XXVIII(1976) , pp . 264-290 . 21. "Heegaar d splitting s o f prime 3-manifold s ar e not unique, " (wit h J.M. Mon - tesinos an d F . Gonzales-Acuna) , Mich. J. Math., 23(1976) , pp. 97-103 . 22. "O n the /x-invariant o f Z-homolog y spheres" , (wit h R . Craggs) , Bull. AMS, 82(1976), pp. 253-255 . 23. "Th e algebrai c structur e o f surface mappin g clas s groups" , Discrete groups and automorphic forms (ed . W . Harvey), Academic Press 1977 , pp. 163-198 . 24. "Th e //-invarian t o f 3-manifold s an d certai n structura l propertie s o f th e group of homeomorphisms o f a closed, oriented 2-manifold", (wit h R. Craggs), Trans. AMS, 237(1978) , pp . 283-309 . 25. "Specia l Heegaar d splitting s fo r closed , oriente d 3-manifolds" , Topology, 17 (1978), pp. 157-166 . 26. "Specia l representations fo r 3-manifolds" , (wit h Jerome Powell) , in Geomet- ric Topology (ed. J.C . Cantrell) , Academi c Press , 1979 , pp. 23-51 . 27. "O n minima l Heegaar d splittings" , (wit h J.M . Montesinos) , Mich. Math. J., 2 7 (1980) , pp. 47-57 . 28. " A representation theore m fo r flbere d knot s an d thei r monodrom y maps" , in Topology of Low Dimensional Manifolds ed . R . Fenn , Springer-Verla g Lecture Notes , no. 722 , 1981 , pp. 1-8 . 29. "Knotte d periodi c orbits i n dynamical system s I; Lorenz's equations", (wit h R.F. Williams ) Topology, 22(1983), pp. 47-82 . 30. "Knotte d periodi c orbit s i n dynamica l system s II ; Kno t holder s fo r flbere d knots", (wit h R.F . Williams) , Contemporary Mathematics, 2 0 (1983) , pp . 1-60. 31. "Fixe d points o f pseudo-Anosov diffeomorphis m o f surfaces", (wit h M . Kid- well), Advances in Math., 46(1982) , pp. 217-220 . 32. "On e side d Heegaar d splitting s an d homeotop y group s o f surfaces" , (wit h J.H. Rubinstein) , Proc. London Math. Soc. (3), 49 (1984) , p. 517-536 . 33. "Abelia n and solvable subgroups o f mapping clas s groups o f surfaces", (wit h A. Lubotzky an d J . McCarthy) , Duke Math. J., 50(1983) , pp. 1107-1120 . 34. "A n algorith m fo r simpl e curve s o n surfaces" , (wit h C . Series) , J. London Math. Soc. (2), 29(1984) , pp . 331-342 . 35. "Geodesie s wit h bounde d intersectio n numbe r o n surface s ar e sparsel y dis - tributed", (wit h C . Series) , Topology, 24 (1985) , pp. 217-225 . 36. "3-fol d branche d covering s and the mapping clas s group o f a surface", (wit h B. Wajnryb), Springer-Verla g Lectur e Notes , no. 1167 , 1986 , pp. 24-46 . 37. "Mappin g clas s group s o f surfaces" , Contemporar y Mathematics , vol . 78 , Braids (ed . J . Birma n an d A . Libgober), 1988 , pp. 13-43 . 38. "Geodesie s with multiple self-intersections an d symmetries o n Riemann sur - faces", (wit h C . Series), London Math Soc . Lectur e Notes, no. 12 , 1988, pp. 3-11. 39. "Dehn' s algorithm revisited , wit h applications to simple curves on surfaces " (with C . Series) , Annals of Math Studies, 111(1989) , pp . 451-478 . 40. "O n the Jone s polynomia l o f closed 3-braids" , Inven. Math., 81(1985) , pp . 287-294= JOAN S . BIRMAN : PUBLICATION S AN D PH.D . THESE S SUPERVISE D x v

41. "Algebrai c linearit y i n th e mappin g clas s grou p o f a surface" , (wit h C . Se - ries), J. of Pure and Applied Algebra, 5 2 (1988) , pp. 227-275 . 42. "Jones ' braid-pla t formul a an d a ne w surger y triple" , (wit h T . Kanenoba) , Proc. AMS, 102(1988) , pp . 687-695 . 43. "Marko v classe s i n certai n finite quotient s o f Artin's brai d group" , (wit h B . Wajnryb), Israel J. Math., 56(1986) , pp . 160-178 . 44. "Braids , lin k polynomial s an d a ne w algebra" , (wit h H . Wenzl) , Trans. AMS, 313(1989) , pp . 249-273 . 45. "O n th e Wor k o f Vaughan Jones" , Proc . Internationa l Congres s o f Mathe - maticians 1990 , Springer-Verlag , Ne w York , 1992 , pp. 9-18 . 46. " A calculus o n link s i n the 3-sphere" , (wit h W. Menasco) , Knot s '9 0 (ed . A . Kawauchi), W . D e Gruyter , Berli n an d Ne w York , 1992 , pp. 625-631 . 47. "Studyin g links via closed braids I: A finiteness theorem", (wit h W. Menasco) , Pacific J. Math., 154(1992) , pp . 17-36 . 48. "Studyin g link s vi a close d braid s II : On a theorem o f Bennequin", (wit h W . Menasco), Topology and its Applications, 4 0 (1991) , pp . 71-82 . 49. "Studyin g link s vi a close d braid s III : Classifyin g link s whic h ar e close d 3 - braids", (wit h W . Menasco) , Pacific J. Math., 161(1993) , pp . 25-113 . 50. "Studyin g link s vi a close d braid s IV : Close d brai d representative s o f spli t and composit e links" , (wit h W . Menasco) , Invent. Math., 102(1990) , pp . 115-139. 51. "Studyin g link s vi a close d braid s V : Close d brai d representative s o f th e unlink", (wit h W . Menasco) , Trans. AMS, 329(1992) , pp . 585-606 . 52. "Studyin g link s vi a close d braid s VI : A non-finitenes s theorem" , (wit h W . Menasco), Pacific J. Math., 156(1992) , pp . 265-285 . 53. "Specia l positions fo r tssential tori in link complements", (wit h W. Menasco) , Topology, 33(1994) , pp . 525-556 ; Errata, Topology. 54. "Kno t polynomials and Vassiliev's Invariants", (wit h X.-S. Lin), Invent.Math., Ill (1993) , pp . 225-270 . 55. "Ne w point s o f vie w i n kno t theory" , Bull. AMS, 28(1993) , pp . 253 - 287 . 56. "Linea r representation s o f the brai d group" , (wit h D . Lon g an d J . Moody) , Contemporary Mathematics, 169(1994) , pp . 123-132 . 57. "Braide d chor d diagrams" , (wit h R . Trapp) , Journal of Knot Theory and its Ramifications, 7(1998) , pp . 1-22 . 58. "Studyin g surface s vi a close d braids" , (wit h Elizabet h Finkelstein) , Journal of Knot Theory and its Ramifications, 7(1998) , pp . 267-334 . 59. " A ne w approac h t o the wor d an d conjugac y problem s i n the brai d groups" , (with J.S . Le e an d K.H . Ko) , Advances in Mathematics, 139(1998) , pp . 322-353. 60. " A ne w algorith m fo r recognizin g th e unknot" , (wit h M . Hirsch) , Geometry and Topology, 2 (1998) , pp . 178-220 . 61. "Holonomi c an d Legendria n parametrization s o f knots", (wit h N . Wrinkle) , to appea r i n Journal of knot Theory and its Ramifications. 62. "O n transversall y simpl e knots" , (wit h N . Wrinkle) , submitte d fo r publica - tion. 63. "Stabilizatio n i n the brai d groups" , (wit h W. Menasco) , manuscrip t i n draf t form. xvi JOA N S . BIRMAN : PUBLICATION S AN D PH.D . THESE S SUPERVISE D

PhD These s supervised : Richard Fein , Steven s Institute o f Technology, 197 4 Marcello Kupferwasser , Columbi a University , 197 5 Jerome Powell , Columbi a University , 197 8 Jozef Przytycki , Columbi a University , 198 1 John McCarthy , Columbi a University , 198 1 Pei Ju n Xu , Columbi a University , 198 7 Rolland Trapp , Columbi a University , 198 7 Elizabeth Finkelstein , Columbi a University , 199 3 Ted Stanford , Columbi a University , 199 3 Zhong-He Chen , Columbi a University , 199 4 Effie Kalfagianni , Columbi a University , 199 4 Ka Y i Ng, Columbia University , 199 6 Tat San g Fung , Columbi a University , 199 6 Matt Greenwood , Columbi a University , 199 6 Hessam Hamidi-Tehrani , Columbi a University , 199 7

Current Ph.D . student s a t Columbi a University : Tara Brendl e Nathan Broaddu s Clement Rad u Popesc u Nancy Wrinkl e Matt Zinn o Photos This page intentionally left blank PAPERS DEDICATE D T O JOA N S . BIRMA N ••nr

Joan Birman, with Joe Birman an d Steve Sawin in the background.

First Row: Heavy discussion between Joan Birman and Rob Kirby. Second Row: Juliet Morton talking to Mark Kid well; Ted Stanford i s on the left . In the third row are Mike Veve and Tatsuya Tsakamotoi. PHOTO SECTIOi N

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Kunio and Yasui Murasugi an d Joan Birman

Vaughan Jones, Bill Menasco, Joan Birman an d Tara Brendle: assembling for lunch. PAPERS DEDICATE D T O JOA N S . BIRMA N xxi

Joan Birman an d Xiao-Song Lin

Steve Sawin, Vaughan Jones, Sofia Lambropoulou (face d awa y from th e camera), and Effie Kalfagianni . PHOTO SECTIO N

Steve Humphries

Former and current Ph.D. students o f Joan Birman. Starting from left : Rolli, Nancy, Liz, Hessam, Tara, Radu, Joan, Efne, Tat-Sang, Ka-Yi, and Jozef. The second an d third fro m right: Ted and John. Others are Jozef's students .