@article {alex1, AUTHOR = {Alexander, J.W.}, TITLE = {Topological Invariants of Knots and Links}, JOURNAL = {Transactions of A.M.S.}, VOLUME = {30}, YEAR = {1928}, PAGES = {275--306}, } @article {MR0040658, AUTHOR = {Blanchfield, R. C. and Fox, Ralph H.}, TITLE = {Invariants of self-linking}, JOURNAL = {Ann. of Math. (2)}, VOLUME = {53}, YEAR = {1951}, PAGES = {556--564}, MRCLASS = {56.0X}, MRNUMBER = {MR0040658 (12,730b)}, MRREVIEWER = {P. J. Hilton}, } @article {MR0085512, AUTHOR = {Blanchfield, R. C.}, TITLE = {Intersection theory of manifolds with operators with applications to knot theory}, JOURNAL = {Ann. of Math. (2)}, VOLUME = {65}, YEAR = {1957}, PAGES = {340--356}, MRCLASS = {55.0X}, MRNUMBER = {MR0085512 (19,53a)}, MRREVIEWER = {R. H. Fox}, } @article{borodzik, title = {A rho-invariant of iterated torus knots}, author = {Maciej Borodzik}, eprint = {arXiv:math.AT/0906.3660}, } @article{borodzik_signatures_2010, title = {On the signatures of torus knots}, url = {http://arxiv.org/abs/1002.4500}, abstract = {We study properties of the signature function of the torus knot {\$T\_{p,q}\$.} First we provide a very elementary proof of the formula for the integral of the signatures over the circle. We obtain also a closed formula for the {Tristram--Levine} signature of a torus knot in terms of Dedekind sums.}, journal = {1002.4500}, author = {Maciej Borodzik and Krzysztof Oleszkiewicz}, month = feb, year = {2010}, keywords = {{57M25,} Mathematics - Geometric Topology}, } @book{Johnson, AUTHOR = {Boswell, James}, TITLE = {The life of Samuel Johnson}, PAGES = {381}, YEAR = {1791}, } @article{54.0373.01, author="Brauner, K.", title="{Zur Geometrie der Funktionen zweier komplexer Ver\"anderlicher. II: Das Verhalten der Funktionen in der Umgebung ihrer Verzweigungsstellen. III: Klassifikation der Singularit\"aten algebroider Kurven. IV: Die Verzweigungsgruppen.}", language="German", journal="Abhandlungen Hamburg ", volume="6", pages="1-55", year="1928", doi={10.1007/BF02940600}, abstract="{Die Verzweigungsstellen einer algebraischen Funktion von $k$ unabh\"angigen Variablen $x_1,\dots,x_k$ sind die Nullstellen der Diskriminante $D(x_1,\dots,x_k)=0$. Nach {\it Wirtinger} unterscheidet man Verzweigungsstellen erster und zweiter Art je nachdem die ``Diskriminantenmannigfaltigkeit'' in diesem Punkte verzweigt ist oder nicht. Aus der Entwicklung von $z$ f\"ur die Umgebung einer Verzweigungastelle erster Art ergibt sich unmittelbar der zyklische Charakter der zugeh\"origen Verzweigungsgruppe. An Hand zweier Beispiele von Funktionen zweier Ver\"anderlichen zeigt Verf. das durchaus andere Verhalten der Funktionen in der Umgebung einer Verzweigungsstelle zweiter Art und f\"uhrt durch seine Beispiele in seine Methode ein. Mit Hilfe einer stereographischen Projektion werden die Schnittkurven der Diskriminantenmannigfaltigkeiten mit einer Hyperkugel in einen geeigneten dreidimensionalen Raum projiziert. Die stereographischen Bildkurven ergeben sich dabei als einfache oder verkettete Torusknoten oder Schlauchknoten, deren Achsen im allgemeinen wiederum Schlauchknoten sind. Die vorliegenden topologischen Verh\"altnisse, die das Verhalten der Funktion in der Umgebung dieser Stelle charakterisieren, bestimmen sich im allgemeinen aus den Charakteristikenpaaren und Indicespaaren der Diskriminantenmannigfaltigkeiten. Im letzten Teil der Arbeit werden die zu den geschilderten topologischen Verh\"altnissen geh\"origen Verzweigungen untersucht. Die Verzweigungen in den ``H\"ochststellungen''. der Kurve erzeugen die zugeh\"orige Gruppe vollst\"andig. Zwischen diesen bestehen besondere Beziehungen, die sog. Wirtingerschen Relationen, aus denen sich durch Reduktion der einfache symmetrische Aufbau der Verzweigungsgruppe aus wesentlichen Erzeugenden ergibt. Zwischen diesen Erzeugenden, die eine einfache geometrische und funktionentheoretische Bedeutung haben, bestehen bestimmte definierende Relationen, die sich aus den Charakteristikenpaaren ableiten lassen.}", reviewer="{Behnke, H.; Prof. (M\"unster in Westfalen)}", } @incollection {MR520521, AUTHOR = {Casson, Andrew and Gordon, C. McA.}, TITLE = {On slice knots in dimension three}, BOOKTITLE = {Algebraic and geometric topology ({P}roc. {S}ympos. {P}ure {M}ath., {S}tanford {U}niv., {S}tanford, {C}alif., 1976), {P}art 2}, SERIES = {Proc. Sympos. Pure Math., XXXII}, PAGES = {39--53}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, R.I.}, YEAR = {1978}, MRCLASS = {57M25}, MRNUMBER = {MR520521 (81g:57003)}, } @incollection {MR900252, AUTHOR = {Casson, Andrew and Gordon, C. McA.}, TITLE = {Cobordism of classical knots}, BOOKTITLE = {\`A la recherche de la topologie perdue}, SERIES = {Progr. Math.}, VOLUME = {62}, PAGES = {181--199}, NOTE = {With an appendix by P. M. Gilmer}, PUBLISHER = {Birkh\"auser Boston}, ADDRESS = {Boston, MA}, YEAR = {1986}, MRCLASS = {57R90 (57M25)}, MRNUMBER = {MR900252}, } @article {MR2054808, AUTHOR = {Cha, Jae Choon and Livingston, Charles}, TITLE = {Knot signature functions are independent}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {132}, YEAR = {2004}, NUMBER = {9}, PAGES = {2809--2816 (electronic)}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {57M25 (11E39)}, MRNUMBER = {MR2054808 (2005d:57004)}, MRREVIEWER = {Alexander Zvonkin}, } @article {MR2343079, AUTHOR = {Cha, Jae Choon}, TITLE = {The structure of the rational concordance group of knots}, JOURNAL = {Mem. Amer. Math. Soc.}, FJOURNAL = {Memoirs of the American Mathematical Society}, VOLUME = {189}, YEAR = {2007}, NUMBER = {885}, PAGES = {x+95}, ISSN = {0065-9266}, CODEN = {MAMCAU}, MRCLASS = {57M25 (57Q45 57Q60)}, MRNUMBER = {MR2343079 (2009c:57007)}, MRREVIEWER = {Swatee Naik}, } @article{cochran_primary_2009, title = {Primary decomposition and the fractal nature of knot concordance}, url = {http://arxiv.org/abs/0906.1373}, abstract = {For each sequence of polynomials, P=(p\_1(t),p\_2(t),...), we define a characteristic series of groups, called the derived series localized at P. Given a knot K in S{\textasciicircum}3, such a sequence of polynomials arises naturally as the orders of certain submodules of the sequence of higher-order Alexander modules of K. These group series yield new filtrations of the knot concordance group that refine the (n)-solvable filtration of {Cochran-Orr-Teichner.} We show that the quotients of successive terms of these refined filtrations have infinite rank. These results also suggest higher-order analogues of the p(t)-primary decomposition of the algebraic concordance group. We use these techniques to give evidence that the set of smooth concordance classes of knots is a fractal set. We also show that no {Cochran-Orr-Teichner} knot is concordant to any {Cochran-Harvey-Leidy} knot.}, journal = {0906.1373}, author = {Tim D. Cochran and Shelly Harvey and Constance Leidy}, month = jun, year = {2009}, keywords ="{57M25,} {20J05,} Mathematics - Geometric Topology, Mathematics - Group Theory } @article {cohale, AUTHOR = {Cochran, Tim D. and Shelly Harvey and Constance Leidy}, TITLE = {Knot concordance and higher-order Blanchfield duality}, } @article{cohale2, title = {Derivatives of knots and second-order signatures}, author = {Cochran, Tim D. and Shelly Harvey and Constance Leidy}, journal = {Algebraic & Geometric Topology}, volume = {10}, year = {2010}, pages = {739–787}, } @article{math.GT/0206059, title = {{Structure in the classical knot concordance group}}, author = {Tim D. Cochran and Kent E. Orr and Peter Teichner}, } @article{math.GT/9908117, title = {{Knot concordance, Whitney towers and L^2 signatures}}, author = {Tim D. Cochran and Kent E. Orr and Peter Teichner}, howpublished = {Ann. of Math. (2), Vol. 157 (2003), no. 2, 433--519}, } @article{collins_l^2_2010, title = {The L^2 signature of torus knots}, url = {http://arxiv.org/abs/1001.1329}, abstract = {We find a formula for the L2 signature of a (p,q) torus knot, which is the integral of the omega-signatures over the unit circle. We then apply this to a theorem of {Cochran-Orr-Teichner} to prove that the n-twisted doubles of the unknot, for n not 0 or 2, are not slice. This is a new proof of the result first proved by Casson and Gordon.}, journal = {1001.1329}, author = {Julia Collins}, year = {2010}, keywords = {{57M25,} {57M27,} Mathematics - Algebraic Topology, Mathematics - Geometric Topology}, } @book {MR0445489, AUTHOR = {Crowell, Richard H. and Fox, Ralph H.}, TITLE = {Introduction to knot theory}, NOTE = {Reprint of the 1963 original, Graduate Texts in Mathematics, No. 57}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1977}, PAGES = {x+182}, MRCLASS = {55A25}, MRNUMBER = {MR0445489 (56 \#3829)}, } @article {MR1364081, AUTHOR = {Epple, Moritz}, TITLE = {Branch points of algebraic functions and the beginnings of modern knot theory}, JOURNAL = {Historia Math.}, FJOURNAL = {Historia Mathematica}, VOLUME = {22}, YEAR = {1995}, NUMBER = {4}, PAGES = {371--401}, ISSN = {0315-0860}, CODEN = {HIMADS}, MRCLASS = {01A60 (01A55 30-03 57-03 57M25)}, MRNUMBER = {MR1364081 (96k:01016)}, MRREVIEWER = {Vagn Lundsgaard Hansen}, DOI = {10.1006/hmat.1995.1031}, URL = {http://dx.doi.org/10.1006/hmat.1995.1031}, } @incollection {MR1674917, AUTHOR = {Epple, Moritz}, TITLE = {Geometric aspects in the development of knot theory}, BOOKTITLE = {History of topology}, PAGES = {301--357}, PUBLISHER = {North-Holland}, ADDRESS = {Amsterdam}, YEAR = {1999}, MRCLASS = {57-03 (57M25 57M27)}, MRNUMBER = {MR1674917 (2000i:57001)}, MRREVIEWER = {Colin C. Adams}, } @article {MR2089303, AUTHOR = {Epple, Moritz}, TITLE = {Knot invariants in {V}ienna and {P}rinceton during the 1920s: epistemic configurations of mathematical research}, JOURNAL = {Sci. Context}, FJOURNAL = {Science in Context}, VOLUME = {17}, YEAR = {2004}, NUMBER = {1-2}, PAGES = {131--164}, ISSN = {0269-8897}, MRCLASS = {01A70 (57-03 57M27)}, MRNUMBER = {MR2089303 (2005e:01020)}, } @article{MR576871, AUTHOR = {Farber, Michael}, TITLE = {Isotopy types of knots of codimension two}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {261}, YEAR = {1980}, NUMBER = {1}, PAGES = {185--209}, ISSN = {0002-9947}, CODEN = {TAMTAM}, MRCLASS = {57Q45 (55P25)}, MRNUMBER = {MR576871 (81k:57016)}, MRREVIEWER = {C. Kearton}, LOCALPDF = "../../papers/farber6.pdf", } @article {MR718824, AUTHOR = {Farber, Michael}, TITLE = {Classification of simple knots}, JOURNAL = {Uspekhi Mat. Nauk}, FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk}, VOLUME = {38}, YEAR = {1983}, NUMBER = {5(233)}, PAGES = {59--106}, ISSN = {0042-1316}, MRCLASS = {57Q45}, MRNUMBER = {MR718824 (85g:57012)}, MRREVIEWER = {Jonathan A. Hillman}, } @article {MR0040658, AUTHOR = {Blanchfield, R. C. and Fox, Ralph H.}, TITLE = {Invariants of self-linking}, JOURNAL = {Ann. of Math. (2)}, VOLUME = {53}, YEAR = {1951}, PAGES = {556--564}, MRCLASS = {56.0X}, MRNUMBER = {MR0040658 (12,730b)}, MRREVIEWER = {P. J. Hilton}, } @article {MR0211392, AUTHOR = {Fox, Ralph H. and Milnor, John W.}, TITLE = {Singularities of {$2$}-spheres in {$4$}-space and cobordism of knots}, JOURNAL = {Osaka J. Math.}, FJOURNAL = {Osaka Journal of Mathematics}, VOLUME = {3}, YEAR = {1966}, PAGES = {257--267}, ISSN = {0030-6126}, MRCLASS = {55.20 (57.00)}, MRNUMBER = {MR0211392 (35 \#2273)}, MRREVIEWER = {C. H. Giffen}, } @book {MR0445489, AUTHOR = {Crowell, Richard H. and Fox, Ralph H.}, TITLE = {Introduction to knot theory}, NOTE = {Reprint of the 1963 original, Graduate Texts in Mathematics, No. 57}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1977}, PAGES = {x+182}, MRCLASS = {55A25}, MRNUMBER = {MR0445489 (56 \#3829)}, } @article{friedl_survey_2009, title = {A survey of twisted Alexander polynomials}, url = {http://arxiv.org/abs/0905.0591}, abstract = {We give a short introduction to the theory of twisted Alexander polynomials of a 3--manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted Reidemeister torsion. We then give a survey of the many applications of twisted invariants to the study of topological problems. We conclude with a short summary of the theory of higher order Alexander polynomials.}, journal = {0905.0591}, author = {Stefan Friedl and Stefano Vidussi}, month = may, year = {2009}, keywords = {{57M27,} {57M25,} Mathematics - Geometric Topology}, } @article{friedl_twisted_2008, title = {Twisted Alexander polynomials detect fibered 3-manifolds}, url = {http://arxiv.org/abs/0805.1234}, abstract = {A classical result in knot theory says that the Alexander polynomial of a fibered knot is monic and that its degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3-manifolds. In this paper we show that the conditions on twisted Alexander polynomials are not only necessary but also sufficient for a 3-manifold to be fibered. By previous work of the authors this result implies that if a manifold of the form S{\textasciicircum}1 x N{\textasciicircum}3 admits a symplectic structure, then N fibers over S{\textasciicircum}1. In fact we will completely determine the symplectic cone of S{\textasciicircum}1 x N in terms of the fibered faces of the Thurston norm ball of N.}, journal = {0805.1234}, author = {Stefan Friedl and Stefano Vidussi}, month = may, year = {2008}, keywords = {{57M27,} Mathematics - Geometric Topology, Mathematics - Symplectic Geometry}, } @article{math.GT/0505233, title = {{New topologically slice knots}}, author = {Stefan Friedl and Peter Teichner}, eprint = {arXiv:math.GT/0505233}, } @article{../../papers/garoufalid, title = {Does the Jones polynomial determine the signature of a knot?}, url = {http://arxiv.org/abs/math/0310203}, abstract = {The signature function of a knot is a locally constant integer valued function with domain the unit circle. The jumps (i.e., the discontinuities) of the signature function can occur only at the roots of the Alexander polynomial on the unit circle. The latter are important in deforming U(1) representations of knot groups to irreducible {SU(2)} representations. Under the assumption that these roots are simple, we formulate a conjecture that explicitly computes the jumps of the signature function in terms of the Jones polynomial of a knot and its parallels. As evidence, we prove our conjecture for torus knots, and also (using computer calculations) for knots with at most 8 crossings. We also give a formula for the jump function at simple roots in terms of relative signs of Alexander polynomials.}, journal = {math/0310203}, author = {Garoufalidis, Stavros}, month = oct, year = {2003}, keywords ="Mathematics - Geometric Topology } @article {MR1507011, AUTHOR = {Goeritz, Lebrecht}, TITLE = {Die {B}etti'schen {Z}ahlen {D}er {Z}yklischen {U}berlagerungsraume {D}er {K}notenaussenraume}, JOURNAL = {Amer. J. Math.}, FJOURNAL = {American Journal of Mathematics}, VOLUME = {56}, YEAR = {1934}, NUMBER = {1-4}, PAGES = {194--198}, ISSN = {0002-9327}, CODEN = {AJMAAN}, MRCLASS = {Contributed Item}, MRNUMBER = {MR1507011}, DOI = {10.2307/2370923}, URL = {http://dx.doi.org/10.2307/2370923}, } @incollection {MR0380766, AUTHOR = {Goldsmith, Deborah L.}, TITLE = {Symmetric fibered links}, BOOKTITLE = {Knots, groups, and {$3$}-manifolds ({P}apers dedicated to the memory of {R}. {H}. {F}ox)}, PAGES = {3--23. Ann. of Math. Studies, No. 84}, PUBLISHER = {Princeton Univ. Press}, ADDRESS = {Princeton, N.J.}, YEAR = {1975}, MRCLASS = {55A10}, MRNUMBER = {MR0380766 (52 \#1663)}, MRREVIEWER = {J. S. Birman}, } @article {MR0500905, AUTHOR = {Gordon, C. McA. and Litherland, R. A.}, TITLE = {On the signature of a link}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {47}, YEAR = {1978}, NUMBER = {1}, PAGES = {53--69}, ISSN = {0020-9910}, MRCLASS = {55A25}, MRNUMBER = {MR0500905 (58 \#18407)}, MRREVIEWER = {Wilbur Whitten}, } @incollection {MR520521, AUTHOR = {Casson, Andrew and Gordon, C. McA.}, TITLE = {On slice knots in dimension three}, BOOKTITLE = {Algebraic and geometric topology ({P}roc. {S}ympos. {P}ure {M}ath., {S}tanford {U}niv., {S}tanford, {C}alif., 1976), {P}art 2}, SERIES = {Proc. Sympos. Pure Math., XXXII}, PAGES = {39--53}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, R.I.}, YEAR = {1978}, MRCLASS = {57M25}, MRNUMBER = {MR520521 (81g:57003)}, } @article {MR617628, AUTHOR = {Gordon, C. McA. and Litherland, R. A. and Murasugi, Kunio}, TITLE = {Signatures of covering links}, JOURNAL = {Canad. J. Math.}, FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de Math\'ematiques}, VOLUME = {33}, YEAR = {1981}, NUMBER = {2}, PAGES = {381--394}, ISSN = {0008-414X}, CODEN = {CJMAAB}, MRCLASS = {57M25}, MRNUMBER = {MR617628 (83a:57006)}, MRREVIEWER = {J. P. Levine}, } @incollection {MR900251, AUTHOR = {Gordon, C. McA.}, TITLE = {On the {$G$}-signature theorem in dimension four}, BOOKTITLE = {\`A la recherche de la topologie perdue}, SERIES = {Progr. Math.}, VOLUME = {62}, PAGES = {159--180}, PUBLISHER = {Birkh\"auser Boston}, ADDRESS = {Boston, MA}, YEAR = {1986}, MRCLASS = {58G10 (57S17)}, MRNUMBER = {MR900251}, } @incollection {MR900252, AUTHOR = {Casson, Andrew and Gordon, C. McA.}, TITLE = {Cobordism of classical knots}, BOOKTITLE = {\`A la recherche de la topologie perdue}, SERIES = {Progr. Math.}, VOLUME = {62}, PAGES = {181--199}, NOTE = {With an appendix by P. M. Gilmer}, PUBLISHER = {Birkh\"auser Boston}, ADDRESS = {Boston, MA}, YEAR = {1986}, MRCLASS = {57R90 (57M25)}, MRNUMBER = {MR900252}, } @article{cochran_primary_2009, title = {Primary decomposition and the fractal nature of knot concordance}, url = {http://arxiv.org/abs/0906.1373}, abstract = {For each sequence of polynomials, P=(p\_1(t),p\_2(t),...), we define a characteristic series of groups, called the derived series localized at P. Given a knot K in S{\textasciicircum}3, such a sequence of polynomials arises naturally as the orders of certain submodules of the sequence of higher-order Alexander modules of K. These group series yield new filtrations of the knot concordance group that refine the (n)-solvable filtration of {Cochran-Orr-Teichner.} We show that the quotients of successive terms of these refined filtrations have infinite rank. These results also suggest higher-order analogues of the p(t)-primary decomposition of the algebraic concordance group. We use these techniques to give evidence that the set of smooth concordance classes of knots is a fractal set. We also show that no {Cochran-Orr-Teichner} knot is concordant to any {Cochran-Harvey-Leidy} knot.}, journal = {0906.1373}, author = {Tim D. Cochran and Shelly Harvey and Constance Leidy}, month = jun, year = {2009}, keywords ="{57M25,} {20J05,} Mathematics - Geometric Topology, Mathematics - Group Theory } @article {cohale, AUTHOR = {Cochran, Tim D. and Shelly Harvey and Constance Leidy}, TITLE = {Knot concordance and higher-order Blanchfield duality}, } @article{cohale2, title = {Derivatives of knots and second-order signatures}, author = {Cochran, Tim D. and Shelly Harvey and Constance Leidy}, journal = {Algebraic & Geometric Topology}, volume = {10}, year = {2010}, pages = {739–787}, } @book {MR0645392, TITLE = {Knot theory}, SERIES = {Lecture Notes in Mathematics}, VOLUME = {685}, BOOKTITLE = {Proceedings of a {S}eminar held in {P}lans-sur-{B}ex, 1977}, AUTHOR = {Hausmann, J. C.}, PUBLISHER = {Springer-Verlag}, ADDRESS = {Berlin}, YEAR = {1978}, PAGES = {ii+311}, ISBN = {3-540-08952-7}, MRCLASS = {57-06}, MRNUMBER = {MR0645392 (58 \#31084)}, } @article {MR0388373, AUTHOR = {Kauffman, Louis and Taylor, Laurence R.}, TITLE = {Signature of links}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {216}, YEAR = {1976}, PAGES = {351--365}, ISSN = {0002-9947}, MRCLASS = {55A25}, MRNUMBER = {MR0388373 (52 \#9210)}, MRREVIEWER = {Wilbur Whitten}, } @incollection {MR521734, AUTHOR = {Kauffman, Louis}, TITLE = {Signature of branched fibrations}, BOOKTITLE = {Knot theory ({P}roc. {S}em., {P}lans-sur-{B}ex, 1977)}, SERIES = {Lecture Notes in Math.}, VOLUME = {685}, PAGES = {203--217}, PUBLISHER = {Springer}, ADDRESS = {Berlin}, YEAR = {1978}, MRCLASS = {57Q45 (55R55 57R40)}, MRNUMBER = {MR521734 (80f:57008)}, MRREVIEWER = {W. D. Neumann}, } @article{1979kearton, jstor_articletype = {primary_article}, title = {The Milnor Signatures of Compound Knots}, author = {Kearton, C.}, journal = {Proceedings of the American Mathematical Society}, jstor_issuetitle = {}, volume = {76}, number = {1}, jstor_formatteddate = {Aug., 1979}, pages = {157--160}, url = {http://www.jstor.org/stable/2042936}, ISSN = {00029939}, abstract = {The Milnor signatures of a classical knot are related to those of its companions.}, language = {}, year = {1979}, publisher = {American Mathematical Society}, copyright = {Copyright © 1979 American Mathematical Society}, } @incollection {MR1803365, AUTHOR = {Kearton, C.}, TITLE = {Quadratic forms in knot theory}, BOOKTITLE = {Quadratic forms and their applications (Dublin, 1999)}, SERIES = {Contemp. Math.}, VOLUME = {272}, PAGES = {135--154}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {2000}, MRCLASS = {57Q45 (11E12 57M25)}, MRNUMBER = {MR1803365 (2002a:57033)}, MRREVIEWER = {J. P. Levine}, } @article {MR534830, AUTHOR = {Kearton, C.}, TITLE = {Signatures of knots and the free differential calculus}, JOURNAL = {Quart. J. Math. Oxford Ser. (2)}, FJOURNAL = {The Quarterly Journal of Mathematics. Oxford. Second Series}, VOLUME = {30}, YEAR = {1979}, NUMBER = {118}, PAGES = {157--182}, ISSN = {0033-5606}, CODEN = {QJMAAT}, MRCLASS = {57M05}, MRNUMBER = {MR534830 (80h:57002)}, MRREVIEWER = {Kenneth A. Perko, Jr.}, } @incollection {MR0178475, AUTHOR = {Kervaire, M.}, TITLE = {On higher dimensional knots}, BOOKTITLE = {Differential and {C}ombinatorial {T}opology ({A} {S}ymposium in {H}onor of {M}arston {M}orse)}, PAGES = {105--119}, PUBLISHER = {Princeton Univ. Press}, ADDRESS = {Princeton, N.J.}, YEAR = {1965}, MRCLASS = {57.20 (55.20)}, MRNUMBER = {MR0178475 (31 \#2732)}, MRREVIEWER = {J. F. Adams}, } @article {MR0189052, AUTHOR = {Kervaire, M.}, TITLE = {Les n\oe uds de dimensions sup\'erieures}, JOURNAL = {Bull. Soc. Math. France}, VOLUME = {93}, YEAR = {1965}, PAGES = {225--271}, MRCLASS = {57.20}, MRNUMBER = {MR0189052 (32 \#6479)}, MRREVIEWER = {E. H. Brown}, } @incollection {MR0283786, AUTHOR = {Kervaire, M.}, TITLE = {Knot cobordism in codimension two}, BOOKTITLE = {Manifolds--Amsterdam 1970 (Proc. Nuffic Summer School)}, SERIES = {Lecture Notes in Mathematics, Vol. 197}, PAGES = {83--105}, PUBLISHER = {Springer}, ADDRESS = {Berlin}, YEAR = {1971}, MRCLASS = {55.20 (57.00)}, MRNUMBER = {MR0283786 (44 \#1016)}, MRREVIEWER = {J. P. Levine}, } @incollection {MR521731, AUTHOR = {Kervaire, M. and Weber, Claude}, TITLE = {A survey of multidimensional knots}, BOOKTITLE = {Knot theory (Proc. Sem., Plans-sur-Bex, 1977)}, SERIES = {Lecture Notes in Math.}, VOLUME = {685}, PAGES = {61--134}, PUBLISHER = {Springer}, ADDRESS = {Berlin}, YEAR = {1978}, MRCLASS = {57Q45 (32C40)}, MRNUMBER = {MR521731 (80f:57009)}, MRREVIEWER = {Louis H. Kauffman}, } @article {MR2092061, AUTHOR = {Kim, Taehee}, TITLE = {Filtration of the classical knot concordance group and {C}asson-{G}ordon invariants}, JOURNAL = {Math. Proc. Cambridge Philos. Soc.}, FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society}, VOLUME = {137}, YEAR = {2004}, NUMBER = {2}, PAGES = {293--306}, ISSN = {0305-0041}, CODEN = {MPCPCO}, MRCLASS = {57M25 (57M27)}, MRNUMBER = {MR2092061 (2005f:57014)}, MRREVIEWER = {Charles Livingston}, } @article {MR0056607, AUTHOR = {Kneser, Martin and Puppe, Dieter}, TITLE = {Quadratische {F}ormen und {V}erschlingungsinvarianten von {K}noten}, JOURNAL = {Math. Z.}, VOLUME = {58}, YEAR = {1953}, PAGES = {376--384}, MRCLASS = {20.0X}, MRNUMBER = {MR0056607 (15,100c)}, MRREVIEWER = {R. H. Fox}, } @article {MR0062438, AUTHOR = {Kyle, R. H.}, TITLE = {Branched covering spaces and the quadratic forms of links}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {59}, YEAR = {1954}, PAGES = {539--548}, ISSN = {0003-486X}, MRCLASS = {56.0X}, MRNUMBER = {MR0062438 (15,979a)}, MRREVIEWER = {R. Bott}, } @article {MR0107240, AUTHOR = {Kyle, R. H.}, TITLE = {Branched covering spaces and the quadratic forms of links. {II}}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {69}, YEAR = {1959}, PAGES = {686--699}, ISSN = {0003-486X}, MRCLASS = {55.00}, MRNUMBER = {MR0107240 (21 \#5965)}, MRREVIEWER = {S. S. Cairns}, } @article{cochran_primary_2009, title = {Primary decomposition and the fractal nature of knot concordance}, url = {http://arxiv.org/abs/0906.1373}, abstract = {For each sequence of polynomials, P=(p\_1(t),p\_2(t),...), we define a characteristic series of groups, called the derived series localized at P. Given a knot K in S{\textasciicircum}3, such a sequence of polynomials arises naturally as the orders of certain submodules of the sequence of higher-order Alexander modules of K. These group series yield new filtrations of the knot concordance group that refine the (n)-solvable filtration of {Cochran-Orr-Teichner.} We show that the quotients of successive terms of these refined filtrations have infinite rank. These results also suggest higher-order analogues of the p(t)-primary decomposition of the algebraic concordance group. We use these techniques to give evidence that the set of smooth concordance classes of knots is a fractal set. We also show that no {Cochran-Orr-Teichner} knot is concordant to any {Cochran-Harvey-Leidy} knot.}, journal = {0906.1373}, author = {Tim D. Cochran and Shelly Harvey and Constance Leidy}, month = jun, year = {2009}, keywords ="{57M25,} {20J05,} Mathematics - Geometric Topology, Mathematics - Group Theory } @article {cohale, AUTHOR = {Cochran, Tim D. and Shelly Harvey and Constance Leidy}, TITLE = {Knot concordance and higher-order Blanchfield duality}, } @article{cohale2, title = {Derivatives of knots and second-order signatures}, author = {Cochran, Tim D. and Shelly Harvey and Constance Leidy}, journal = {Algebraic & Geometric Topology}, volume = {10}, year = {2010}, pages = {739–787}, } @article {MR0246314, AUTHOR = {Levine, J.}, TITLE = {Knot cobordism groups in codimension two}, JOURNAL = {Comment. Math. Helv.}, VOLUME = {44}, YEAR = {1969}, PAGES = {229--244}, MRCLASS = {57.20}, MRNUMBER = {MR0246314 (39 \#7618)}, MRREVIEWER = {R. Schultz}, } @article {MR0253348, AUTHOR = {Levine, J.}, TITLE = {Invariants of knot cobordism}, JOURNAL = {Invent. Math. 8 (1969), 98--110; addendum, ibid.}, VOLUME = {8}, YEAR = {1969}, PAGES = {355}, MRCLASS = {57.10}, MRNUMBER = {MR0253348 (40 \#6563)}, MRREVIEWER = {R. H. Fox}, } @article {MR1004605, AUTHOR = {Levine, J.}, TITLE = {Metabolic and hyperbolic forms from knot theory}, JOURNAL = {J. Pure Appl. Algebra}, FJOURNAL = {Journal of Pure and Applied Algebra}, VOLUME = {58}, YEAR = {1989}, NUMBER = {3}, PAGES = {251--260}, ISSN = {0022-4049}, CODEN = {JPAAA2}, MRCLASS = {57Q45 (11E39)}, MRNUMBER = {MR1004605 (90h:57027)}, MRREVIEWER = {C. Kearton}, DOI = {10.1016/0022-4049(89)90040-6}, URL = {http://dx.doi.org/10.1016/0022-4049(89)90040-6}, } @book {MR1472978, AUTHOR = {Lickorish, W. B. R.}, TITLE = {An introduction to knot theory}, SERIES = {Graduate Texts in Mathematics}, VOLUME = {175}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1997}, PAGES = {x+201}, ISBN = {0-387-98254-X}, MRCLASS = {57M25 (57N10)}, MRNUMBER = {MR1472978 (98f:57015)}, MRREVIEWER = {Darryl McCullough}, } @article {MR0500905, AUTHOR = {Gordon, C. McA. and Litherland, R. A.}, TITLE = {On the signature of a link}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {47}, YEAR = {1978}, NUMBER = {1}, PAGES = {53--69}, ISSN = {0020-9910}, MRCLASS = {55A25}, MRNUMBER = {MR0500905 (58 \#18407)}, MRREVIEWER = {Wilbur Whitten}, } @incollection {MR547456, AUTHOR = {Litherland, R. A.}, TITLE = {Signatures of iterated torus knots}, BOOKTITLE = {Topology of low-dimensional manifolds ({P}roc. {S}econd {S}ussex {C}onf., {C}helwood {G}ate, 1977)}, SERIES = {Lecture Notes in Math.}, VOLUME = {722}, PAGES = {71--84}, PUBLISHER = {Springer}, ADDRESS = {Berlin}, YEAR = {1979}, MRCLASS = {57M25}, MRNUMBER = {MR547456 (80k:57012)}, MRREVIEWER = {Lee Rudolph}, } @article {MR617628, AUTHOR = {Gordon, C. McA. and Litherland, R. A. and Murasugi, Kunio}, TITLE = {Signatures of covering links}, JOURNAL = {Canad. J. Math.}, FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de Math\'ematiques}, VOLUME = {33}, YEAR = {1981}, NUMBER = {2}, PAGES = {381--394}, ISSN = {0008-414X}, CODEN = {CJMAAB}, MRCLASS = {57M25}, MRNUMBER = {MR617628 (83a:57006)}, MRREVIEWER = {J. P. Levine}, } @incollection {MR780587, AUTHOR = {Litherland, R. A.}, TITLE = {Cobordism of satellite knots}, BOOKTITLE = {Four-manifold theory ({D}urham, {N}.{H}., 1982)}, SERIES = {Contemp. Math.}, VOLUME = {35}, PAGES = {327--362}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {1984}, MRCLASS = {57M25 (18F25 19G12 19M05 57Q60)}, MRNUMBER = {MR780587 (86k:57003)}, MRREVIEWER = {Cameron McA. Gordon}, } @article {MR1879982, AUTHOR = {Livingston, Charles}, TITLE = {New examples of non-slice, algebraically slice knots}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {130}, YEAR = {2002}, NUMBER = {5}, PAGES = {1551--1555 (electronic)}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {57M25 (57M12)}, MRNUMBER = {MR1879982 (2002j:57018)}, MRREVIEWER = {Daniel Matignon}, DOI = {10.1090/S0002-9939-01-06201-3}, URL = {http://dx.doi.org/10.1090/S0002-9939-01-06201-3}, } @article {MR2054808, AUTHOR = {Cha, Jae Choon and Livingston, Charles}, TITLE = {Knot signature functions are independent}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {132}, YEAR = {2004}, NUMBER = {9}, PAGES = {2809--2816 (electronic)}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {57M25 (11E39)}, MRNUMBER = {MR2054808 (2005d:57004)}, MRREVIEWER = {Alexander Zvonkin}, } @article{livingston_algebraic_2008, title = {The algebraic concordance order of a knot}, url = {http://arxiv.org/abs/0806.3068}, abstract = {Levine defined the rational algebraic knot concordance group and proved that each nontrivial element is of order two, of order four, or of infinite order. The determination of the order of an element depends on a p-adic analysis for all primes p. Here we develop effective means to determine the order of any element that is in the image of the integral algebraic concordance group by restricting the set of primes that need to be considered and by finding simple tests that often avoid p-adic considerations. The paper includes an outline of how the results apply to give the determination of the algebraic orders of all 2,977 prime knots of 12 or fewer crossings. The paper also includes a short expository account of the necessary background in p-adic numbers and Witt groups of bilinear forms.}, journal = {0806.3068}, author = {Charles Livingston}, month = jun, year = {2008}, keywords = {{57M25,} Mathematics - Geometric Topology}, } @article {MR0437456, AUTHOR = {Matumoto, Takao}, TITLE = {On the signature invariants of a non-singular complex sesquilinear form}, JOURNAL = {J. Math. Soc. Japan}, VOLUME = {29}, YEAR = {1977}, NUMBER = {1}, PAGES = {67--71}, MRCLASS = {10C05 (57D20 57C45)}, MRNUMBER = {MR0437456 (55 \#10386)}, MRREVIEWER = {F. Hirzebruch}, } @article {MR0211392, AUTHOR = {Fox, Ralph H. and Milnor, John W.}, TITLE = {Singularities of {$2$}-spheres in {$4$}-space and cobordism of knots}, JOURNAL = {Osaka J. Math.}, FJOURNAL = {Osaka Journal of Mathematics}, VOLUME = {3}, YEAR = {1966}, PAGES = {257--267}, ISSN = {0030-6126}, MRCLASS = {55.20 (57.00)}, MRNUMBER = {MR0211392 (35 \#2273)}, MRREVIEWER = {C. H. Giffen}, } @incollection {MR0242163, AUTHOR = {Milnor, John W.}, TITLE = {Infinite cyclic coverings}, BOOKTITLE = {Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967)}, PAGES = {115--133}, PUBLISHER = {Prindle, Weber \& Schmidt, Boston, Mass.}, YEAR = {1968}, MRCLASS = {57.01 (55.00)}, MRNUMBER = {MR0242163 (39 \#3497)}, MRREVIEWER = {M. A. Kervaire}, } @article {MR0249519, AUTHOR = {Milnor, John W.}, TITLE = {On isometries of inner product spaces}, JOURNAL = {Invent. Math.}, VOLUME = {8}, YEAR = {1969}, PAGES = {83--97}, MRCLASS = {20.75 (15.00)}, MRNUMBER = {MR0249519 (40 \#2764)}, MRREVIEWER = {S. B{\"o}ge}, } @article {MR0171275, AUTHOR = {Murasugi, Kunio}, TITLE = {On a certain numerical invariant of link types}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {117}, YEAR = {1965}, PAGES = {387--422}, ISSN = {0002-9947}, MRCLASS = {55.20}, MRNUMBER = {MR0171275 (30 \#1506)}, MRREVIEWER = {R. H. Fox}, } @incollection {MR2276141, AUTHOR = {Murasugi, Kunio}, TITLE = {Classical knot invariants and elementary number theory}, BOOKTITLE = {Primes and knots}, SERIES = {Contemp. Math.}, VOLUME = {416}, PAGES = {167--196}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {2006}, MRCLASS = {57M27 (11Z05 57M25)}, MRNUMBER = {MR2276141 (2009b:57028)}, MRREVIEWER = {Ying Zhang}, } @book {MR2347576, AUTHOR = {Murasugi, Kunio}, TITLE = {Knot theory \& its applications}, SERIES = {Modern Birkh\"auser Classics}, NOTE = {Translated from the 1993 Japanese original by Bohdan Kurpita, Reprint of the 1996 translation [MR1391727]}, PUBLISHER = {Birkh\"auser Boston Inc.}, ADDRESS = {Boston, MA}, YEAR = {2008}, PAGES = {x+341}, ISBN = {978-0-8176-4718-6}, MRCLASS = {57M25}, MRNUMBER = {MR2347576}, } @article {MR617628, AUTHOR = {Gordon, C. McA. and Litherland, R. A. and Murasugi, Kunio}, TITLE = {Signatures of covering links}, JOURNAL = {Canad. J. Math.}, FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de Math\'ematiques}, VOLUME = {33}, YEAR = {1981}, NUMBER = {2}, PAGES = {381--394}, ISSN = {0008-414X}, CODEN = {CJMAAB}, MRCLASS = {57M25}, MRNUMBER = {MR617628 (83a:57006)}, MRREVIEWER = {J. P. Levine}, } @article{borodzik_signatures_2010, title = {On the signatures of torus knots}, url = {http://arxiv.org/abs/1002.4500}, abstract = {We study properties of the signature function of the torus knot {\$T\_{p,q}\$.} First we provide a very elementary proof of the formula for the integral of the signatures over the circle. We obtain also a closed formula for the {Tristram--Levine} signature of a torus knot in terms of Dedekind sums.}, journal = {1002.4500}, author = {Maciej Borodzik and Krzysztof Oleszkiewicz}, month = feb, year = {2010}, keywords = {{57M25,} Mathematics - Geometric Topology}, } @article{math.GT/0206059, title = {{Structure in the classical knot concordance group}}, author = {Tim D. Cochran and Kent E. Orr and Peter Teichner}, } @article{math.GT/9908117, title = {{Knot concordance, Whitney towers and L^2 signatures}}, author = {Tim D. Cochran and Kent E. Orr and Peter Teichner}, howpublished = {Ann. of Math. (2), Vol. 157 (2003), no. 2, 433--519}, } @article {slice, AUTHOR = {Powell, Mark}, TITLE = {Notes of Peter Teichner's 2001 San Diego course}, } @article {MR0056607, AUTHOR = {Kneser, Martin and Puppe, Dieter}, TITLE = {Quadratische {F}ormen und {V}erschlingungsinvarianten von {K}noten}, JOURNAL = {Math. Z.}, VOLUME = {58}, YEAR = {1953}, PAGES = {376--384}, MRCLASS = {20.0X}, MRNUMBER = {MR0056607 (15,100c)}, MRREVIEWER = {R. H. Fox}, } @article {MR2058802, AUTHOR = {Ranicki, Andrew}, TITLE = {Blanchfield and {S}eifert algebra in high-dimensional knot theory}, JOURNAL = {Mosc. Math. J.}, FJOURNAL = {Moscow Mathematical Journal}, VOLUME = {3}, YEAR = {2003}, NUMBER = {4}, PAGES = {1333--1367}, ISSN = {1609-3321}, MRCLASS = {19J25 (57Q45)}, MRNUMBER = {MR2058802 (2005b:19008)}, MRREVIEWER = {Masayuki Yamasaki}, } @book{ranickibook, title="High-dimensional knot theory", author="Andrew Ranicki", publisher="Springer", year="1998", } @paper{ranickislides, title="On the signatures of knots", author="Andrew Ranicki", notes={Slides of lecture, Durham, June 2010}, } @article {MR0182965, AUTHOR = {Robertello, Raymond A.}, TITLE = {An invariant of knot cobordism}, JOURNAL = {Comm. Pure Appl. Math.}, FJOURNAL = {Communications on Pure and Applied Mathematics}, VOLUME = {18}, YEAR = {1965}, PAGES = {543--555}, ISSN = {0010-3640}, MRCLASS = {55.20}, MRNUMBER = {MR0182965 (32 \#447)}, MRREVIEWER = {A. Haefliger}, } @article{0008.18101, author="Seifert, H.", title="{Verschlingungsinvarianten.}", language="German", year="1933", } @article{61.0609.03, author="Seifert, H.", title="{Die Verschlingungsinvarianten der zyklischen Knoten\"uberlagerungen.}", language="German", journal="Abhandl. Hamburg ", volume="11", pages="84-101", year="1935", } @article {MR0275415, AUTHOR = {Shinohara, Yaichi}, TITLE = {On the signature of knots and links}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {156}, YEAR = {1971}, PAGES = {273--285}, ISSN = {0002-9947}, MRCLASS = {55.20}, MRNUMBER = {MR0275415 (43 \#1172)}, MRREVIEWER = {H. E. Debrunner}, } @book{Silver, AUTHOR = {Silver, Daniel}, TITLE = {Scottish Physics and Knot Theory's Odd Origins}, YEAR = {2005}, } @article {MR0467764, AUTHOR = {Stoltzfus, Neal W.}, TITLE = {Unraveling the integral knot concordance group}, JOURNAL = {Mem. Amer. Math. Soc.}, FJOURNAL = {Memoirs of the American Mathematical Society}, VOLUME = {12}, YEAR = {1977}, NUMBER = {192}, PAGES = {iv+91}, ISSN = {0065-9266}, MRCLASS = {57C45 (10C05)}, MRNUMBER = {MR0467764 (57 \#7616)}, MRREVIEWER = {J. P. Levine}, } @article {MR0388373, AUTHOR = {Kauffman, Louis and Taylor, Laurence R.}, TITLE = {Signature of links}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {216}, YEAR = {1976}, PAGES = {351--365}, ISSN = {0002-9947}, MRCLASS = {55A25}, MRNUMBER = {MR0388373 (52 \#9210)}, MRREVIEWER = {Wilbur Whitten}, } @article{math.GT/0206059, title = {{Structure in the classical knot concordance group}}, author = {Tim D. Cochran and Kent E. Orr and Peter Teichner}, } @article{math.GT/0304299, title = {{Knots, von Neumann signatures, and grope cobordism}}, author = {Peter Teichner}, howpublished = {Proceedings of the ICM, Beijing 2002, vol. 2, 437--446}, } @article{math.GT/0505233, title = {{New topologically slice knots}}, author = {Stefan Friedl and Peter Teichner}, eprint = {arXiv:math.GT/0505233}, } @article{math.GT/9908117, title = {{Knot concordance, Whitney towers and L^2 signatures}}, author = {Tim D. Cochran and Kent E. Orr and Peter Teichner}, howpublished = {Ann. of Math. (2), Vol. 157 (2003), no. 2, 433--519}, } @article {MR0248854, AUTHOR = {Tristram, A. G.}, TITLE = {Some cobordism invariants for links}, JOURNAL = {Proc. Cambridge Philos. Soc.}, VOLUME = {66}, YEAR = {1969}, PAGES = {251--264}, MRCLASS = {57.01 (55.00)}, MRNUMBER = {MR0248854 (40 \#2104)}, MRREVIEWER = {H. E. Debrunner}, } @article{cot, title = {Notes on Cochran-Orr-Teichner, Muenster 2004}, author = {Marco Varisco}, } @article{friedl_survey_2009, title = {A survey of twisted Alexander polynomials}, url = {http://arxiv.org/abs/0905.0591}, abstract = {We give a short introduction to the theory of twisted Alexander polynomials of a 3--manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted Reidemeister torsion. We then give a survey of the many applications of twisted invariants to the study of topological problems. We conclude with a short summary of the theory of higher order Alexander polynomials.}, journal = {0905.0591}, author = {Stefan Friedl and Stefano Vidussi}, month = may, year = {2009}, keywords = {{57M27,} {57M25,} Mathematics - Geometric Topology}, } @article{friedl_twisted_2008, title = {Twisted Alexander polynomials detect fibered 3-manifolds}, url = {http://arxiv.org/abs/0805.1234}, abstract = {A classical result in knot theory says that the Alexander polynomial of a fibered knot is monic and that its degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3-manifolds. In this paper we show that the conditions on twisted Alexander polynomials are not only necessary but also sufficient for a 3-manifold to be fibered. By previous work of the authors this result implies that if a manifold of the form S{\textasciicircum}1 x N{\textasciicircum}3 admits a symplectic structure, then N fibers over S{\textasciicircum}1. In fact we will completely determine the symplectic cone of S{\textasciicircum}1 x N in terms of the fibered faces of the Thurston norm ball of N.}, journal = {0805.1234}, author = {Stefan Friedl and Stefano Vidussi}, month = may, year = {2008}, keywords = {{57M27,} Mathematics - Geometric Topology, Mathematics - Symplectic Geometry}, } @article {MR0370605, AUTHOR = {Viro, O. Ya.}, TITLE = {Branched coverings of manifolds with boundary, and invariants of links. {I}}, JOURNAL = {Izv. Akad. Nauk SSSR Ser. Mat.}, FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya}, VOLUME = {37}, YEAR = {1973}, PAGES = {1241--1258}, ISSN = {0373-2436}, MRCLASS = {57C45}, MRNUMBER = {MR0370605 (51 \#6832)}, MRREVIEWER = {O. Krotenheerdt}, } @article {MR771233, AUTHOR = {Viro, O. Ya.}, TITLE = {The signature of a branched covering}, JOURNAL = {Mat. Zametki}, FJOURNAL = {Akademiya Nauk SSSR. Matematicheskie Zametki}, VOLUME = {36}, YEAR = {1984}, NUMBER = {4}, PAGES = {549--557}, ISSN = {0025-567X}, MRCLASS = {57M12 (57R20)}, MRNUMBER = {MR771233 (87d:57004)}, MRREVIEWER = {Jo{\v{z}}e Vrabec}, } @incollection {MR521731, AUTHOR = {Kervaire, M. and Weber, Claude}, TITLE = {A survey of multidimensional knots}, BOOKTITLE = {Knot theory (Proc. Sem., Plans-sur-Bex, 1977)}, SERIES = {Lecture Notes in Math.}, VOLUME = {685}, PAGES = {61--134}, PUBLISHER = {Springer}, ADDRESS = {Berlin}, YEAR = {1978}, MRCLASS = {57Q45 (32C40)}, MRNUMBER = {MR521731 (80f:57009)}, MRREVIEWER = {Louis H. Kauffman}, }