Algebraic & Geometric

We apply Kauffman bracket skein algebras to develop a theory of skein adequate links in thickened surfaces. We show that any alternating link diagram on a surface is skein adequate. We apply our theory to establish the first and second Tait conjectures for adequate links in thickened surfaces. Our notion of skein adequacy is broader and more powerful than the corresponding notions of adequacy previously considered for link diagrams in surfaces. For a link diagram D on a surface † of minimal genus g.†/ , we show that

[1]  K. Baker,et al.  Census L–space knots are braid positive, except for one that is not , 2022, Algebraic & Geometric Topology.

[2]  A. Manion.  On the decategorification of some higher actions in Heegaard Floer homology , 2022, Algebraic & Geometric Topology.

[3]  Ramin Naimi,et al.  Intrinsically knotted graphs with linklessly embeddable simple minors , 2021, Algebraic & Geometric Topology.

[4]  Henry Adams,et al.  The persistent topology of optimal transport based metric thickenings , 2021, Algebraic & Geometric Topology.

[5]  Hongjian Yang.  Annular Khovanov homology and augmented links , 2021, Algebraic & Geometric Topology.

[6]  Ahmad Issa,et al.  Equivariantly slicing strongly negative amphichiral knots , 2021, Algebraic & Geometric Topology.

[7]  Guozhen Wang,et al.  Computations of the Adams-Novikov E2-term , 2021, Chinese Annals of Mathematics, Series B.

[8]  Nicholas L. Georgakopoulos.  The RO(C4) cohomology of the infinite real projective space , 2021, Algebraic & Geometric Topology.

[9]  Dylan Wilson,et al.  On the $C_p$-equivariant dual Steenrod algebra , 2021, 2103.16006.

[10]  Elena Pavelescu,et al.  Constructions stemming from nonseparating planar graphs and their Colin de Verdière invariant , 2021, Algebraic & Geometric Topology.

[11]  H. Boden,et al.  Minimal crossing number implies minimal supporting genus , 2020, Bulletin of the London Mathematical Society.

[12]  M. Scharlemann.  Generating the Goeritz group of $S^3$ , 2020, 2011.10613.

[13]  M. Behrens,et al.  The 2–primary Hurewicz image of tmf , 2020, Geometry & Topology.

[14]  Juan Luis Dur'an Batalla,et al.  The deformation space of nonorientable hyperbolic 3–manifolds , 2020, Algebraic & Geometric Topology.

[15]  H. Boden,et al.  A characterization of alternating links in thickened surfaces , 2020, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[16]  Stefan Friedl.  Algebraic topology , 2020, Graduate Studies in Mathematics.

[17]  Abhijit Champanerkar,et al.  A volumish theorem for alternating virtual links , 2020, 2010.08499.

[18]  Efstratia Kalfagianni,et al.  Guts, volume and skein modules of 3-manifolds , 2020, Fundamenta Mathematicae.

[19]  Joshua Wang.  Link Floer homology also detects split links , 2020, Bulletin of the London Mathematical Society.

[20]  M. Bendersky,et al.  A Cartan formula for the cohomology of polyhedral products and its application to the ring structure , 2020, 2009.06818.

[21]  N. Petrosyan,et al.  Cohomological and geometric invariants of simple complexes of groups , 2020, 2009.02161.

[22]  John A. Baldwin,et al.  Khovanov homology detects the figure‐eight knot , 2020, Bulletin of the London Mathematical Society.

[23]  Allison H. Moore,et al.  Triple linking numbers and Heegaard Floer homology , 2020, 2006.15484.

[24]  Zhenkun Li,et al.  Two detection results of Khovanov homology on links , 2020, Transactions of the American Mathematical Society.

[25]  Gage Martin.  Khovanov homology detects $T(2,6)$ , 2020, Mathematical Research Letters.

[26]  Sanath K. Devalapurkar.  Higher chromatic Thom spectra via unstable homotopy theory , 2020, Algebraic & Geometric Topology.

[27]  Daniel Isaksen,et al.  Stable homotopy groups of spheres: from dimension 0 to 90 , 2020, Publications mathématiques de l'IHÉS.

[28]  Yi Wang,et al.  Turaev hyperbolicity of classical and virtual knots , 2019, Algebraic & Geometric Topology.

[29]  Nicholas L. Georgakopoulos.  The $RO(C_4)$ integral homology of a point , 2019, 1912.06758.

[30]  Yi Xie,et al.  Classification of links with Khovanov homology of minimal rank , 2019, Journal of the European Mathematical Society.

[31]  H. Boden,et al.  The Jones–Krushkal polynomial and minimal diagrams of surface links , 2019, Annales de l'Institut Fourier.

[32]  Thang T. Q. Lê,et al.  Stated skein algebras of surfaces , 2019, Journal of the European Mathematical Society.

[33]  Yi Xie,et al.  Instanton Floer homology for sutured manifolds with tangles , 2019, 1907.00547.

[34]  Beibei Liu.  L‐space surgeries on 2‐component L‐space links , 2019, Transactions of the London Mathematical Society.

[35]  H. Karimi.  The Khovanov Homology of Alternating Virtual Links , 2019, Michigan Mathematical Journal.

[36]  Ryan Blair,et al.  Twist number and the alternating volume of knots , 2019, Journal of Knot Theory and Its Ramifications.

[37]  Clover May.  A structure theorem for RO(C2)–graded Bredon cohomology , 2018, 1804.03691.

[38]  C. Adams,et al.  Hyperbolicity of links in thickened surfaces , 2018, Topology and its Applications.

[39]  John A. Baldwin,et al.  Khovanov homology detects the trefoils , 2018, Duke Mathematical Journal.

[40]  John A. Baldwin,et al.  A note on the knot Floer homology of fibered knots , 2018, Algebraic & Geometric Topology.

[41]  Jessica S. Purcell,et al.  Geometry of alternating links on surfaces , 2017, Transactions of the American Mathematical Society.

[42]  Thang T. Q. Lê,et al.  Unicity for representations of the Kauffman bracket skein algebra , 2017, Inventiones mathematicae.

[43]  Donald Yau,et al.  Smith ideals of operadic algebras in monoidal model categories , 2017, Algebraic & Geometric Topology.

[44]  F. Cohen,et al.  A spectral sequence for polyhedral products , 2015, 1511.08292.

[45]  A. Saltz,et al.  An annular refinement of the transverse element in Khovanov homology , 2015, 1507.06263.

[46]  Xiangjun Wang,et al.  The homology of simplicial complements and the cohomology of polyhedral products , 2015 .

[47]  Anthony M. Licata,et al.  Annular Khovanov homology and knotted Schur–Weyl representations , 2015, Compositio Mathematica.

[48]  Diana Hubbard.  On Sutured Khovanov Homology and Axis-Preserving Mutations , 2015, 1505.03091.

[49]  M. Hopkins,et al.  The slice spectral sequence for the $C_{4}$ analog of real $K$-theory , 2015, 1502.07611.

[50]  E. Gorsky,et al.  Cable links and L-space surgeries , 2015, 1502.05425.

[51]  Margaret I. Doig,et al.  On the intersection ring of graph manifolds , 2014, 1412.3990.

[52]  L. Cai.  On products in a real moment-angle manifold , 2014, 1410.5543.

[53]  Naoki Kitazawa.  Fold maps with singular value sets of concentric spheres , 2014 .

[54]  M. Masuda,et al.  Cohomology of toric origami manifolds with acyclic proper faces , 2014, 1407.0764.

[55]  Li Yu.  On Hochster's formula for a class of quotient spaces of moment-angle complexes , 2014, 1406.7392.

[56]  Abhijit Champanerkar,et al.  A Survey on the Turaev Genus of Knots , 2014, 1406.1945.

[57]  M. Shulman.  Equivariant local coefficients and the RO(G)-graded cohomology of classifying spaces , 2014, 1405.1770.

[58]  M. Hedden,et al.  On the geography and botany of knot Floer homology , 2014, Selecta Mathematica.

[59]  K. Ormsby,et al.  Galois equivariance and stable motivic homotopy theory , 2014, 1401.4728.

[60]  V. Manturov.  PARITY AND PROJECTION FROM VIRTUAL KNOTS TO CLASSICAL KNOTS , 2013 .

[61]  Naoki Kitazawa.  Constructions of round fold maps on smooth bundles , 2013, 1307.7270.

[62]  John A. Baldwin,et al.  Categorified invariants and the braid group , 2012, 1212.2222.

[63]  A. R. Pires,et al.  The topology of toric origami manifolds , 2012, 1211.6435.

[64]  Keiko Kawamuro,et al.  Essential open book foliations and fractional Dehn twist coefficient , 2012, 1208.1559.

[65]  F. Cohen,et al.  Cup-products for the polyhedral product functor , 2012, Mathematical Proceedings of the Cambridge Philosophical Society.

[66]  M. Masuda.  Toric topology , 2012, 1203.4399.

[67]  V. Buchstaber,et al.  Nerve complexes and moment-angle spaces of convex polytopes , 2011 .

[68]  Mainak Poddar,et al.  A class of torus manifolds with nonconvex orbit space , 2011, 1109.0798.

[69]  P. B. Kronheimer,et al.  Khovanov homology is an unknot-detector , 2010, 1005.4346.

[70]  K. Shimomura.  Note on beta elements in homotopy, and an application to the prime three case , 2010 .

[71]  F. Bonahon,et al.  Quantum traces for representations of surface groups in SL2(ℂ) , 2010, 1003.5250.

[72]  Zhi Lü,et al.  Moment-angle complexes from simplicial posets , 2009, 0912.2219.

[73]  V. Guillemin,et al.  Symplectic Origami , 2009, 0909.4065.

[74]  M. Hopkins,et al.  On the nonexistence of elements of Kervaire invariant one , 2009, 0908.3724.

[75]  J. E. Grigsby,et al.  Khovanov Homology, Sutured Floer Homology, and Annular Links , 2009, 0907.4375.

[76]  Vyacheslav Krushkal,et al.  Graphs, Links, and Duality on Surfaces , 2009, Combinatorics, Probability and Computing.

[77]  M. Lackenby.  The crossing number of composite knots , 2008, 0805.4706.

[78]  P. Ozsváth,et al.  Holomorphic disks, link invariants and the multi-variable Alexander polynomial , 2008 .

[79]  P. Goerss.  The Adams-Novikov Spectral Sequence and the Homotopy Groups of Spheres , 2008, 0802.1006.

[80]  F. Cohen,et al.  The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces , 2007, 0711.4689.

[81]  Takahiko Yoshida.  Local torus actions modeled on the standard representation , 2007, 0710.2166.

[82]  Lawrence P. Roberts.  On knot Floer homology in double branched covers , 2007, 0706.0741.

[83]  Bruce W. Westbury,et al.  Confluence Theory for Graphs , 2006, math/0609832.

[84]  D. Thurston,et al.  Cluster algebras and triangulated surfaces. Part I: Cluster complexes , 2006, math/0608367.

[85]  Ryan Blair.  ALTERNATING AUGMENTATIONS OF LINKS , 2006, math/0608405.

[86]  Oliver T. Dasbach,et al.  The Jones polynomial and graphs on surfaces , 2006, J. Comb. Theory B.

[87]  P. Ghiggini.  Knot Floer homology detects genus-one fibred knots , 2006, math/0603445.

[88]  P. Ozsváth,et al.  Link Floer homology and the Thurston norm , 2006, math/0601618.

[89]  Alina Iacob,et al.  Generalized Tate cohomology , 2005 .

[90]  Y. Diao.  THE ADDITIVITY OF CROSSING NUMBERS , 2004 .

[91]  Adam S. Sikora,et al.  Categorification of the Kauffman bracket skein module of I-bundles over surfaces , 2004, math/0409414.

[92]  Thang T. Q. Lê.  The colored Jones polynomial and the A-polynomial of Knots☆ , 2004, math/0407521.

[93]  L. Kauffman,et al.  Minimal surface representations of virtual knots and links , 2004, math/0401035.

[94]  Tilman Bauer.  Computation of the homotopy of the spectrum tmf , 2003, math/0311328.

[95]  A. Goncharov,et al.  Moduli spaces of local systems and higher Teichmüller theory , 2003, math/0311149.

[96]  T. Panov,et al.  On the cohomology of torus manifolds , 2003, math/0306100.

[97]  J. González-Meneses.  The nth root of a braid is unique up to conjugacy , 2003, math/0306070.

[98]  Emmanuel Giroux.  Géométrie de contact: de la dimension trois vers les dimensions supérieures , 2003, math/0305129.

[99]  M. Behrens,et al.  On the existence of the self map v_2^9 on the Smith-Toda complex V(1) at the prime 3 , 2003, math/0303223.

[100]  G. Kuperberg.  What is a virtual link , 2002, math/0208039.

[101]  Naoko Kamada.  On the Jones polynomials of checkerboard colorable virtual links , 2002 .

[102]  Victor Matveevich Buchstaber,et al.  Torus Actions and their Applications in Topology and Combinatorics , 2002 .

[103]  Igor Kriz,et al.  Real-oriented homotopy theory and an analogue of the Adams–Novikov spectral sequence , 2001 .

[104]  M. Lackenby.  The Volume of Hyperbolic Alternating Link Complements , 2000, math/0012185.

[105]  M. Saito,et al.  STABLE EQUIVALENCE OF KNOTS ON SURFACES AND VIRTUAL KNOT COBORDISMS , 2000, math/0008118.

[106]  Martin Scharlemann,et al.  Chapter 18 – Heegaard Splittings of Compact 3-Manifolds , 2000, math/0007144.

[107]  S. Kamada,et al.  ABSTRACT LINK DIAGRAMS AND VIRTUAL KNOTS , 2000 .

[108]  Volkmar Welker,et al.  Homotopy colimits – comparison lemmas for combinatorial applications , 1999 .

[109]  J. Przytycki.  Fundamentals of Kauffman bracket skein modules , 1998, math/9809113.

[110]  Adam S. Sikora,et al.  On skein algebras and Sl2(C)-character varieties , 1997, q-alg/9705011.

[111]  J. Hass,et al.  Neon Bulbs and the Unknotting of ARCS in Manifolds , 1997 .

[112]  D. Bullock,et al.  UNDERSTANDING THE KAUFFMAN BRACKET SKEIN MODULE , 1996, q-alg/9604013.

[113]  N. Habegger,et al.  Topological Auantum Field Theories derived from the Kauffman bracket , 1995 .

[114]  R. Stong.  The Jones polynomial of parallels and applications to crossing number , 1994 .

[115]  M. Thistlethwaite,et al.  The classification of alternating links , 1993 .

[116]  O. Saeki.  SIMPLE STABLE MAPS OF 3-MANIFOLDS INTO SURFACES(Real Singularities and Real Algebraic Geometry) , 1992 .

[117]  V. Turaev.  Algebras of Loops on Surfaces, Algebras of Knots, and Quantization , 1991 .

[118]  T. Januszkiewicz,et al.  Convex polytopes, Coxeter orbifolds and torus actions , 1991 .

[119]  V. Turaev.  a Simple Proof of the Murasugi and Kauffman Theorems on Alternating Links , 1990 .

[120]  T. Lê.  On Segre products of affine semigroup rings , 1988 .

[121]  Michaela Davis.  The homology of a space on which a reflection group acts , 1987 .

[122]  D. Ravenel.  Complex Cobordism and Stable Homotopy Groups of Spheres , 1986 .

[123]  D. Eisenbud,et al.  Three-Dimensional Link Theory and Invariants of Plane Curve Singularities. , 1986 .

[124]  J. Hoste.  The first coefficient of the Conway polynomial , 1985 .

[125]  Michael W. Davis.  Groups Generated by reflections and aspherical manifolds not covered by Euclidean space , 1983 .

[126]  Shichirô Oka.  Note on the β-family in stable homotopy of spheres at the prime 3 , 1981 .

[127]  W. Neumann.  A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves , 1981 .

[128]  D. Ravenel.  The non-existence of odd primary Arf invariant elements in stable homotopy , 1978, Mathematical Proceedings of the Cambridge Philosophical Society.

[129]  Robert Myers.  Open book decompositions of 3-manifolds , 1978 .

[130]  D. Ravenel,et al.  Periodic phenomena in the Adams-Novikov spectral sequence , 1977 .

[131]  H. Toda.  On spectra realizing exterior parts of the steenrod algebra , 1971 .

[132]  H. Levine.  Elimination of cusps , 1965 .

[133]  W. Chow.  On Unmixedness Theorem , 1964 .

[134]  H. Whitney.  On Singularities of Mappings of Euclidean Spaces. I. Mappings of the Plane Into the Plane , 1955 .

[135]  I. James.  REDUCED PRODUCT SPACES , 1955 .

[136]  O. Saeki,et al.  Round fold maps of n-dimensional manifolds into (n-1)-dimensional Euclidean space , 2023, Journal of Singularities.

[137]  G. Muller.  Skein and cluster algebras of marked surfaces , 2016 .

[138]  Then,et al.  THE BETA ELEMENTS βtp2/r IN THE HOMOTOPY OF SPHERES , 2009 .

[139]  H. A..  THE HOMOTOPY GROUPS OF tmf AND OF ITS LOCALIZATIONS , 2009 .

[140]  E. Riehl.  HOMOTOPY (LIMITS AND) COLIMITS , 2009 .

[141]  Michael Davis,et al.  The geometry and topology of Coxeter groups , 2008 .

[142]  Igor Frenkel,et al.  A Categorification of the Jones Polynomial , 2008 .

[143]  M. Franz.  The integral cohomology of toric manifolds , 2006 .

[144]  Osamu Saeki,et al.  Topology of Singular Fibers of Differentiable Maps , 2004, Lecture notes in mathematics.

[145]  J. Bryden,et al.  A Linking Form Conjecture for 3-manifolds , 2004 .

[146]  B. Sturmfels,et al.  Combinatorial Commutative Algebra , 2004 .

[147]  L. Kauffman.  An Introduction to Knot Theory , 2001 .

[148]  J. Palmieri.  Stable homotopy over the Steenrod algebra , 2001 .

[149]  W. S. Massey,et al.  THE COHOMOLOGY STRUCTURE OF CERTAIN FlBRE SPACES-I , 2001 .

[150]  K. Shimomura.  THE HOMOTOPY GROUPS OF THE L2-LOCALIZED TODA-SMITH SPECTRUM V (1) AT THE PRIME 3 , 1997 .

[151]  M. Scharlemann,et al.  Thin position and Heegaard splittings of the 3-sphere , 1994 .

[152]  O. Saeki.  Simple stable maps of $3$-manifolds into surfaces II , 1993 .

[153]  L. T. Hoa,et al.  Segre products and rees algebras of face rings , 1992 .

[154]  V. Turaev.  Skein quantization of Poisson algebras of loops on surfaces , 1991 .

[155]  Louis H. Kauffman,et al.  State Models and the Jones Polynomial , 1987 .

[156]  Morwen Thistlethwaite,et al.  A spanning tree expansion of the jones polynomial , 1987 .

[157]  Harold Levine,et al.  Classifying immersions into IR[4] over stable maps of 3-manifolds into IR[2] , 1985 .

[158]  W. Menasco.  Closed incompressible surfaces in alternating knot and link complements , 1984 .

[159]  F. Johnson.  On the triangulation of stratified sets and singular varieties , 1983 .

[160]  K. Murasugi.  On closed 3-braids , 1974 .

[161]  Walter D. Neumann,et al.  Differentiable manifolds and quadratic forms , 1971 .

[162]  G. Whitehead.  Homotopy groups of joins and unions , 1956 .