A sheaf-theoretic \(\mathrm{SL}(2, \mathbb{C})\) Floer homology for knots. (English) Zbl 1482.57010
The paper extends the construction of the sheaf-theoretic \(\textrm{SL}(2,{\mathbb C})\) Floer cohomology of \(3\)-manifolds defined by M. Abouzaid and C. Manolescu [J. Eur. Math. Soc. (JEMS) 22, No. 11, 3641–3695 (2020; Zbl 1455.57018)] to the setting of knots in \(3\)-manifolds. The development of \(\textrm{SL}(2,{\mathbb C})\) Floer cohomology of \(3\)-manifolds, and of its extension to knots constructed here, is motivated in part by the approach of E. Witten [Quantum Topol. 3, No. 1, 1–137 (2012; Zbl 1241.57041)] to Khovanov homology via gauge theory.
This invariant is a sheaf-theoretic \(\textrm{SL}(2,{\mathbb C})\) analogue of the instanton Floer homology of P. B. Kronheimer and T. S. Mrowka [J. Topol. 4, No. 4, 835–918 (2011; Zbl 1302.57064)] defined using \(\textrm{SU}(2)\) connections singular along a knot, with asymptotically traceless holonomy around the meridian. Starting from a Heegaard splitting where the Heegaard surface intersects the knot in two points, the authors consider the relative character variety, consisting of irreducible \(\textrm{SL}(2,{\mathbb C})\) connections on the Heegaard surface with holonomy trace given by a fixed value in \((-2,2)\) around the two points. The two handlebodies of the Heegaard decomposition give rise to a pair of complex Lagrangians, and the invariant is defined using the theory of perverse sheaves of vanishing cycles on their intersection, extending the construction of Abouzaid and Manolescu.
In certain examples, the two-bridge and torus knots, it is shown that the \(\textrm{SL}(2,{\mathbb C})\) invariant is determined by the \(l\)-degree of the \(\hat A\)-polynomial, but this is not true in general.
This invariant is a sheaf-theoretic \(\textrm{SL}(2,{\mathbb C})\) analogue of the instanton Floer homology of P. B. Kronheimer and T. S. Mrowka [J. Topol. 4, No. 4, 835–918 (2011; Zbl 1302.57064)] defined using \(\textrm{SU}(2)\) connections singular along a knot, with asymptotically traceless holonomy around the meridian. Starting from a Heegaard splitting where the Heegaard surface intersects the knot in two points, the authors consider the relative character variety, consisting of irreducible \(\textrm{SL}(2,{\mathbb C})\) connections on the Heegaard surface with holonomy trace given by a fixed value in \((-2,2)\) around the two points. The two handlebodies of the Heegaard decomposition give rise to a pair of complex Lagrangians, and the invariant is defined using the theory of perverse sheaves of vanishing cycles on their intersection, extending the construction of Abouzaid and Manolescu.
In certain examples, the two-bridge and torus knots, it is shown that the \(\textrm{SL}(2,{\mathbb C})\) invariant is determined by the \(l\)-degree of the \(\hat A\)-polynomial, but this is not true in general.
Reviewer: Slava Krushkal (Charlottesville)
MSC:
| 57K18 | Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) |
| 57R58 | Floer homology |
| 57K31 | Invariants of 3-manifolds (including skein modules, character varieties) |
Keywords:
knots in 3-manifolds; Floer homology; \(\mathrm{SL}(2, \mathbb{C})\) connections; perverse sheavesReferences:
| [1] | M.Abouzaid and C.Manolescu, ‘A sheaf‐theoretic model for \(\operatorname{SL} ( 2 , \mathbb{C} )\) Floer homology’, Preprint, 2017, arXiv:1708.00289 [math.GT]. |
| [2] | G.Arone and M.Kankaanrinta, ‘On the functoriality of the blow‐up construction’, Bull. Belg. Math. Soc. Simon Stevin17 (2010) 821-832. · Zbl 1252.57011 |
| [3] | I.Biswas and K.Guruprasad, ‘Principal bundles on open surfaces and invariant functions on Lie groups’, Int. J. Math.4 (1993) 535-544. · Zbl 0789.58017 |
| [4] | H. U.Boden and C. L.Curtis, ‘The \(\operatorname{ SL } ( 2 , \operatorname{C} )\) Casson invariant for knots and the \(\hat{A} \)‐polynomial’, Canad. J. Math.68 (2016) 3-23. · Zbl 1335.57018 |
| [5] | H. U.Boden and K.Yokogawa, ‘Moduli spaces of parabolic Higgs bundles and parabolic \(K ( D )\) pairs over smooth curves. I’, Internat. J. Math.7 (1996) 573-598. · Zbl 0883.14012 |
| [6] | S.Boyer, E.Luft and X.Zhang, ‘On the algebraic components of the \(\operatorname{ SL } ( 2 , \mathbb{C} )\) character varieties of knot exteriors’, Topology41 (2002) 667-694. · Zbl 0998.57017 |
| [7] | S.Boyer and X.Zhang, ‘A proof of the finite filling conjecture’, J. Differential Geom.59 (2001) 87-176. · Zbl 1030.57024 |
| [8] | K. S.Brown, Cohomology of groups, Graduate Texts in Mathematics 87 (Springer, New York, NY, 1994). (Corrected reprint of the 1982 original.) |
| [9] | R.Budney, ‘Looking for large knot examples’, MathOverflow version 2013‐09‐04, https://mathoverflow.net/q/141251. |
| [10] | G.Burde, ‘\( \operatorname{ SU } ( 2 )\)‐representation spaces for two‐bridge knot groups’, Math. Ann.288 (1990) 103-119. · Zbl 0694.57003 |
| [11] | V.Bussi, ‘Categorification of Lagrangian intersections on complex symplectic manifolds using perverse sheaves of vanishing cycles’, Preprint, 2014, arXiv:1404.1329 [math.AG]. |
| [12] | R.Cohen, ‘The topology of fiber bundles’, Lecture notes, http://math.stanford.edu/ ralph/fiber.pdf. |
| [13] | D.Cooper, M.Culler, H.Gillet, D. D.Long and P. B.Shalen, ‘Plane curves associated to character varieties of 3‐manifolds’, Invent. Math.118 (1994) 47-84. · Zbl 0842.57013 |
| [14] | D.Cooper and D. D.Long, ‘Remarks on the \(A\)‐polynomial of a knot’, J. Knot Theory Ramifications5 (1996) 609-628. · Zbl 0890.57012 |
| [15] | M.Culler and P. B.Shalen, ‘Varieties of group representations and splittings of 3‐manifolds’, Ann. of Math. (2) 117 (1983) 109-146. · Zbl 0529.57005 |
| [16] | S. K.Donaldson, Floer homology groups in Yang‐Mills theory, Cambridge Tracts in Mathematics 147 (Cambridge University Press, Cambridge, 2002). (With the assistance of M. Furuta and D. Kotschick.) · Zbl 0998.53057 |
| [17] | J.‐M.Drézet, ‘Luna’s slice theorem and applications’, Algebraic group actions and quotients (ed. J. A.Wisniewski (ed.); Hindawi Publishing Corporation, Cairo, 2004) 39-89. · Zbl 1109.14307 |
| [18] | N. M.Dunfield, ‘Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds’, Invent. Math.136 (1999) 623-657. · Zbl 0928.57012 |
| [19] | A.Floer, ‘An instanton‐invariant for 3‐manifolds’, Comm. Math. Phys.118 (1988) 215-240. · Zbl 0684.53027 |
| [20] | A.Floer, ‘Instanton homology, surgery, and knots’, Geometry of low‐dimensional manifolds, London Mathematical Society Lecture Note Series 150 (eds S. K.Donaldson (ed.) and C. B.Thomas (ed.); Cambridge University Press, Cambridge, 1990) 97-114. · Zbl 0788.57008 |
| [21] | O.García‐Prada, P. B.Gothen and V.Muñoz, Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, Memoirs of the American Mathematical Society 187 (American Mathematical Society, Providence, RI, 2007). · Zbl 1132.14031 |
| [22] | S.Garoufalidis and T. W.Mattman, ‘The \(A\)‐polynomial of the \(( - 2 , 3 , 3 + 2 n )\) pretzel knots’, New York J. Math.17 (2011) 269-279. · Zbl 1233.57002 |
| [23] | W. M.Goldman, ‘The complex‐symplectic geometry of \(\operatorname{ SL } ( 2 , \mathbb{C} )\)‐characters over surfaces’, Algebraic groups and arithmetic (eds S. G.Dani (ed.) and G.Prasad (ed.); Tata Institute of Fundamental Research, Mumbai, 2004) 375-407. · Zbl 1089.53060 |
| [24] | M.Gulbrandsen, Local aspects of geometric invariant theory, https://people.kth.se/ gulbr/local‐git.pdf. |
| [25] | R.Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52 (Springer, New York‐Heidelberg, 1977). · Zbl 0367.14001 |
| [26] | A.Hatcher and W.Thurston, ‘Incompressible surfaces in 2‐bridge knot complements’, Invent. Math.79 (1985) 225-246. · Zbl 0602.57002 |
| [27] | C. M.Herald, ‘Flat connections, the Alexander invariant, and Casson’s invariant’, Comm. Anal. Geom.5 (1997) 93-120. · Zbl 0893.57003 |
| [28] | H.Hida, Elementary theory of \(L\)‐functions and Eisenstein series, London Mathematical Society Student Texts 26 (Cambridge University Press, Cambridge, 1993). · Zbl 0942.11024 |
| [29] | D.Joyce, ‘A classical model for derived critical loci’, J. Differential Geom.101 (2015) 289-367. · Zbl 1368.14027 |
| [30] | A.Juhász, ‘Holomorphic discs and sutured manifolds’, Algebr. Geom. Topol.6 (2006) 1429-1457. · Zbl 1129.57039 |
| [31] | A.Juhász, D.Thurston and I.Zemke, ‘Naturality and mapping class groups in Heegaard Floer homology’, Preprint, 2012, arXiv:1210.4996v4, 2018. · Zbl 1510.57001 |
| [32] | M.Kapovich, ‘On character varieties of 3‐manifold groups’, 2015, http://sgt.imj‐prg.fr/curve2015/Slides/Kapovich.pdf. (Talk slides.) |
| [33] | H.Konno, ‘Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface’, J. Math. Soc. Japan45 (1993) 253-276. · Zbl 0787.53022 |
| [34] | P. B.Kronheimer and T. S.Mrowka, ‘Khovanov homology is an unknot‐detector’, Publ. Math. Inst. Hautes Études Sci.113 (2011) 97-208. · Zbl 1241.57017 |
| [35] | P. B.Kronheimer and T. S.Mrowka, ‘Knot homology groups from instantons’, J. Topol.4 (2011) 835-918. · Zbl 1302.57064 |
| [36] | P. B.Kronheimer and T. S.Mrowka, ‘Gauge theory and Rasmussen’s invariant’, J. Topol.6 (2013) 659-674. · Zbl 1298.57025 |
| [37] | F.Laudenbach ‘A proof of Reidemeister‐Singer’s theorem by Cerf’s methods’, Ann. Fac. Sci. Toulouse Math. (6) 23 (2014) 197-221. (English, with English and French summaries.) · Zbl 1322.57020 |
| [38] | T. T. Q.Le and A. T.Tran, ‘On the AJ conjecture for knots’, Indiana Univ. Math. J.64 (2015) 1103-1151. (With an appendix written jointly with Vu Q. Huynh.) · Zbl 1329.57011 |
| [39] | J. M.Lee, Introduction to smooth manifolds, 2nd edn, Graduate Texts in Mathematics 218 (Springer, New York, NY, 2013). · Zbl 1258.53002 |
| [40] | X.‐S.Lin, ‘A knot invariant via representation spaces’, J. Differential Geom.35 (1992) 337-357. · Zbl 0774.57007 |
| [41] | A.Lubotzky and R. A.Magid, Varieties of representations of finitely generated groups, Memoirs of the American Math Society 58 (American Mathematical Society, Providence, RI, 1985). · Zbl 0598.14042 |
| [42] | M. L.Macasieb and T. W.Mattman, ‘Commensurability classes of \(( - 2 , 3 , n )\) pretzel knot complements’, Algebr. Geom. Topol.8 (2008) 1833-1853. · Zbl 1162.57005 |
| [43] | J.Marché, ‘Character varieties in \(\operatorname{SL}_2\) and Kauffman skein algebras’, Preprint, 2015, arXiv:1510.09107. |
| [44] | Y.Manin, Introduction to the theory of schemes, Moscow Lectures (Springer, New York, NY, 1975). |
| [45] | D. V.Mathews, ‘An explicit formula for the \(A\)‐polynomial of twist knots’, J. Knot Theory Ramifications23 (2014) 1450044, https://doi.org/10.1142/S0218216514500448. · Zbl 1302.57043 · doi:10.1142/S0218216514500448 |
| [46] | T.Mattman, ‘The Culler‐Shalen seminorms of Pretzel knots’, PhD Thesis, McGill University, 2000. |
| [47] | J. W.Milnor and J. D.Stasheff, Characteristic classes, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 76. · Zbl 0298.57008 |
| [48] | G.Mondello, ‘Topology of representation spaces of surface groups in \(\operatorname{ PSL }_2 ( \mathbb{R} )\) with assigned boundary monodromy and nonzero Euler number’, Pure Appl. Math. Q.12 (2016) 399-462. · Zbl 1394.30030 |
| [49] | S.Mukai, An introduction to invariants and moduli, Cambridge Studies in Advanced Mathematics 81 (Cambridge University Press, Cambridge, 2003). (Translated from the 1998 and 2000 Japanese editions by W. M. Oxbury.) · Zbl 1033.14008 |
| [50] | M.Ozawa, ‘Knots and surfaces’, Preprint, 2016, arXiv:1603.09039. |
| [51] | J.Porti, ‘Varieties of characters and knot symmetries (I)’, http://mat.uab.es/ |
| [52] | A. H. W.Schmitt, Geometric invariant theory and decorated principal bundles, Zurich Lectures in Advanced Mathematics (European Mathematical Society, Zürich, 2008). · Zbl 1159.14001 |
| [53] | P. B.Shalen, ‘Representations of 3‐manifold groups’, Handbook of geometric topology (eds R. J.Daverman (ed.) and R. B.Sher (ed.); North‐Holland, Amsterdam, 2002) 955-1044. · Zbl 1012.57003 |
| [54] | A. S.Sikora, ‘Character varieties’, Trans. Am. Math. Soc.364 (2012) 5173-5208. · Zbl 1291.14022 |
| [55] | C.Simpson, ‘Harmonic bundles on noncompact curves’, J. Amer. Math. Soc.3 (1990) 713-770. · Zbl 0713.58012 |
| [56] | M.Thaddeus, ‘Variation of moduli of parabolic Higgs bundles’, J. reine angew. Math.547 (2002) 1-14. · Zbl 0995.14011 |
| [57] | C. M.Tsau, ‘Incompressible surfaces in the knot manifolds of torus knots’, Topology33 (1994) 197-201. · Zbl 0841.57027 |
| [58] | R.Vakil, ‘The rising sea: foundations of algebraic geometry notes, September 2015 version’, http://math.stanford.edu/ vakil/216blog/index.html. |
| [59] | F. W.Warner, Foundations of differentiable manifolds and lie groups, Graduate Texts in Mathematics (Springer, New York, NY, 1983). · Zbl 0516.58001 |
| [60] | A.Weil, ‘Remarks on the cohomology of groups’, Ann. of Math.80 (1964) 149-158. · Zbl 0192.12802 |
| [61] | E.Witten, ‘Fivebranes and knots’, Quantum Topol.3 (2012) 1-137. · Zbl 1241.57041 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
