{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:16:02Z","timestamp":1760242562578,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2017,10,17]],"date-time":"2017-10-17T00:00:00Z","timestamp":1508198400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of     X =  O  P 1    ( \u2212 1 , \u2212 1 )      to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero, one boundary component open Gromov-Witten invariants for Lagrangian submanifolds      L K  \u2282 X     obtained from the conormal bundles of knots     K \u2282  S 3     . This computation is then performed for two non-toric examples (the figure-eight and three-twist knots). For     ( r , s )     torus knots, the open Gromov-Witten invariants can also be computed using Atiyah-Bott localization. Using this result for the unknot and the     ( 3 , 2 )     torus knot, we show that the augmentation polynomial can be derived from these open Gromov-Witten invariants.<\/jats:p>","DOI":"10.3390\/sym9100232","type":"journal-article","created":{"date-parts":[[2017,10,17]],"date-time":"2017-10-17T11:14:35Z","timestamp":1508238875000},"page":"232","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Open Gromov-Witten Invariants from the Augmentation Polynomial"],"prefix":"10.3390","volume":"9","author":[{"given":"Matthew","family":"Mahowald","sequence":"first","affiliation":[{"name":"Department of Mathematics, Northwestern University, Evanston, IL 60208, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,10,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/0550-3213(91)90292-6","article-title":"A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory","volume":"359","author":"Candelas","year":"1991","journal-title":"Nuclear Phys. B"},{"key":"ref_2","unstructured":"Witten, E. (1992). Mirror manifolds and topological field theory. Essays on Mirror Manifolds, International Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1415","DOI":"10.4310\/ATMP.1999.v3.n5.a5","article-title":"On the gauge theory\/geometry correspondence","volume":"3","author":"Gopakumar","year":"1999","journal-title":"Adv. Theor. Math. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1016\/S0550-3213(00)00118-8","article-title":"Knot invariants and topological strings","volume":"577","author":"Ooguri","year":"2000","journal-title":"Nuclear Phys. B"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/978-1-4612-4264-2_12","article-title":"Enumeration of rational curves via torus actions","volume":"Volume 129","author":"Kontsevich","year":"1995","journal-title":"The Moduli Space of Curves (Texel Island, 1994)"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1007\/s002220050293","article-title":"Localization of virtual classes","volume":"135","author":"Graber","year":"1999","journal-title":"Invent. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"495","DOI":"10.4310\/ATMP.1999.v3.n3.a3","article-title":"Local mirror symmetry: Calculations and interpretations","volume":"3","author":"Chiang","year":"1999","journal-title":"Adv. Theor. Math. Phys."},{"key":"ref_8","unstructured":"Klemm, A., and Zaslow, E. (arXiv, 1999). Local mirror symmetry at higher genus, arXiv."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1007\/BF01217730","article-title":"Quantum field theory and the Jones polynomial","volume":"121","author":"Witten","year":"1989","journal-title":"Commun. Math. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1007\/s00220-012-1563-3","article-title":"Large N duality, Lagrangian cycles, and algebraic knots","volume":"319","author":"Diaconescu","year":"2013","journal-title":"Commun. Math. Phys."},{"key":"ref_11","unstructured":"Aganagic, M., and Vafa, C. (arXiv, 2012). Large N duality, mirror symmetry, and a Q-deformed A-polynomial for knots, arXiv."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"827","DOI":"10.4310\/ATMP.2014.v18.n4.a3","article-title":"Topological strings, D-model, and knot contact homology","volume":"18","author":"Aganagic","year":"2014","journal-title":"Adv. Theor. Math. Phys."},{"key":"ref_13","unstructured":"Ng, L. (arXiv, 2012). A topological introduction to knot contact homology, arXiv."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/0550-3213(96)00434-8","article-title":"Mirror symmetry is T-duality","volume":"479","author":"Strominger","year":"1996","journal-title":"Nuclear Phys. B"},{"key":"ref_15","unstructured":"Aganagic, M., and Vafa, C. (arXiv, 2000). Mirror symmetry, D-branes and counting holomorphic discs, arXiv."},{"key":"ref_16","unstructured":"Hori, K., and Vafa, C. (arXiv, 2000). Mirror symmetry, arXiv."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/zna-2002-9-1001","article-title":"Disk instantons, mirror symmetry and the duality web","volume":"57","author":"Aganagic","year":"2002","journal-title":"Z. Naturforsch. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4310\/ATMP.2001.v5.n1.a1","article-title":"Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc","volume":"5","author":"Katz","year":"2001","journal-title":"Adv. Theor. Math. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Labastida, J., Mari\u00f1o, M., and Vafa, C. (2001). Knots, links and branes at large N. J. High Energy Phys., 2000.","DOI":"10.1088\/1126-6708\/2000\/11\/007"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1873","DOI":"10.1007\/s00023-012-0171-2","article-title":"Torus knots and mirror symmetry","volume":"13","author":"Brini","year":"2012","journal-title":"Ann. Henri Poincar\u00e9"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1007\/s11005-014-0687-0","article-title":"Torus knots and the topological vertex","volume":"104","author":"Jockers","year":"2014","journal-title":"Lett. Math. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"975","DOI":"10.2140\/gt.2013.17.975","article-title":"Knot contact homology","volume":"17","author":"Ekholm","year":"2013","journal-title":"Geom. Topol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"987","DOI":"10.1007\/s00220-014-2238-z","article-title":"Knot invariants from topological recursion on augmentation varieties","volume":"336","author":"Gu","year":"2015","journal-title":"Commun. Math. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Neitzke, A., and Walcher, J. (arXiv, 2007). background independence and the open topological string wavefunction, arXiv.","DOI":"10.1090\/pspum\/078\/2483755"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1090\/conm\/310\/05400","article-title":"Open-string Gromov-Witten invariants: calculations and a mirror \u201ctheorem\u201d","volume":"Volume 310","author":"Graber","year":"2002","journal-title":"Orbifolds in Mathematics and Physics (Madison, WI, 2001)"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1155\/S1073792896000414","article-title":"Equivariant Gromov-Witten invariants","volume":"1996","author":"Givental","year":"1996","journal-title":"Internat. Math. Res. Notices"},{"key":"ref_27","unstructured":"Brini, A., and Cavalieri, R. (arXiv, 2010). Open orbifold Gromov\u2013Witten invariants of [\n              \n                \n                  \n                    \n                      C\n                      2\n                    \n\t\t\t\t\t\/\n\t\t\t\t\t \n                      Z\n                      n\n                    \n                  \n                \n              \n            ]: Localization and Mirror Symmetry, arXiv."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1007\/s00220-012-1489-9","article-title":"Open topological strings and integrable hierarchies: remodeling the A-model","volume":"312","author":"Brini","year":"2012","journal-title":"Commun. Math. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Cox, D.A., and Katz, S. (1999). Mirror symmetry and algebraic geometry. Mathematical Surveys and Monographs, American Mathematical Society.","DOI":"10.1090\/surv\/068"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1215\/00127094-2009-015","article-title":"Computing genus-zero twisted Gromov-Witten invariants","volume":"147","author":"Coates","year":"2009","journal-title":"Duke Math. J."},{"key":"ref_31","unstructured":"Brini, A. Personal communication."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/10\/232\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:47:37Z","timestamp":1760208457000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/10\/232"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,17]]},"references-count":31,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2017,10]]}},"alternative-id":["sym9100232"],"URL":"https:\/\/doi.org\/10.3390\/sym9100232","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2017,10,17]]}}}