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12 results on '"Shende V"'

1. Cluster varieties from legendrian knots

Author

Shende, V, Shende, V, Treumann, D, Williams, H, Zaslow, E, Shende, V, Shende, V, Treumann, D, Williams, H, and Zaslow, E

Abstract

Many interesting spaces-including all positroid strata and wild character varieties- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact Lagrangian fillings of Legendrian links in the standard contact three space.

Published
2019

2. Cluster varieties from legendrian knots

Author

Shende, V, Shende, V, Treumann, D, Williams, H, Zaslow, E, Shende, V, Shende, V, Treumann, D, Williams, H, and Zaslow, E

Abstract

Many interesting spaces-including all positroid strata and wild character varieties- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact Lagrangian fillings of Legendrian links in the standard contact three space.

Published
2019

3. The hilbert scheme of a plane curve singularity and the HOMFLY homology of its link

Author

Oblomkov, A, Oblomkov, A, Rasmussen, J, Shende, V, Gorsky, E, Oblomkov, A, Oblomkov, A, Rasmussen, J, Shende, V, and Gorsky, E

Abstract

We conjecture an expression for the dimensions of the Khovanov-Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture (2012) relating the HOMFLY polynomial to the Euler numbers of the same spaces upon setting t=-1. By generalizing results of Piontkowski on the structure of compactified Jacobians to the case of Hilbert schemes of points, we give an explicit prediction of the HOMFLY homology of a (k,n) torus knot as a certain sum over diagrams. The Hilbert scheme series corresponding to the summand of the HOMFLY homology with minimal “a” grading can be recovered from the perverse filtration on the cohomology of the compactified Jacobian. In the case of (k,n) torus knots, this space furnishes the unique finite-dimensional simple representation of the rational spherical Cherednik algebra with central character k/n. Up to a conjectural identification of the perverse filtration with a previously introduced filtration, the work of Haiman and Gordon and Stafford gives formulas for the Hilbert scheme series when k=mn+1.

Published
2018

4. The hilbert scheme of a plane curve singularity and the HOMFLY homology of its link

Author

Oblomkov, A, Oblomkov, A, Rasmussen, J, Shende, V, Gorsky, E, Oblomkov, A, Oblomkov, A, Rasmussen, J, Shende, V, and Gorsky, E

Abstract

We conjecture an expression for the dimensions of the Khovanov-Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture (2012) relating the HOMFLY polynomial to the Euler numbers of the same spaces upon setting t=-1. By generalizing results of Piontkowski on the structure of compactified Jacobians to the case of Hilbert schemes of points, we give an explicit prediction of the HOMFLY homology of a (k,n) torus knot as a certain sum over diagrams. The Hilbert scheme series corresponding to the summand of the HOMFLY homology with minimal “a” grading can be recovered from the perverse filtration on the cohomology of the compactified Jacobian. In the case of (k,n) torus knots, this space furnishes the unique finite-dimensional simple representation of the rational spherical Cherednik algebra with central character k/n. Up to a conjectural identification of the perverse filtration with a previously introduced filtration, the work of Haiman and Gordon and Stafford gives formulas for the Hilbert scheme series when k=mn+1.

Published
2018

5. Torus knots and the rational DAHA

Author

Gorsky, E, Gorsky, E, Oblomkov, A, Rasmussen, J, Shende, V, Gorsky, E, Gorsky, E, Oblomkov, A, Rasmussen, J, and Shende, V

Abstract

We conjecturally extract the triply graded Khovanov-Rozansky homology of the (m;n) torus knot from the unique finite-dimensional simple representation of the rational DAHA of type A, rank n - 1, and central character m/n. The conjectural differentials of Gukov, Dunfield, and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov-Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to q; t-Catalan numbers and to previous conjectures of the last three authors relating knot homology to Hilbert schemes on singular curves.

Published
2014

6. Heparin-induced thrombocytopenia in the critically ill: Interpreting the 4Ts test in a randomized trial.

Author

Hebert P., Murphy E., Foxall J., Skrobik Y., Harvey J., Chitu S., Sirois C., Nadon C., Dolle S., Barge D., Caf T., Howe B., Bellomo R., Eastwood G., Peck L., Goldsmith D., O'Sullivan K., Ernest D., Radford S., Whitfield A., Cross A., Eliott S., Sidhu J., Mercer I., Hamilton A., Botha J., Vuat J., Allsop S., Fowler N., Crozier T., Barrett J., Wright C., Galt P., Culhane C., Ioannidis R., Burton S., Reily M., Weeraratna C., Seppelt I., Weisbrodt L., Bond R., Evans D., Rivett J., O'Connor S., Poole A., Woolfe C., Rajbhandari D., Rees C., Edington J., Fletcher J., Smith J., Boschert C., Reece G., Sara T., Nand K., Bersten A., Gallus A., Matheson E., O'Callaghan M., Orford N., Elderkin T., Fraser M., Bone A., Salerno T., Kinmonth A., Arora S., O'Bree B., Shepherd K., Davey-Quinn A., Sterba M., Johnson B.R., Xu R., Hill F., Ramadoss R., Wood J., Crowther M., Cook D., Guyatt G., Zytaruk N., McDonald E., Williamson D., Albert M., Dodek P., Finfer S., Vallance S., Heels-Ansdell D., McIntyre L., Mehta S., Lamontagne F., Muscedere J., Jacka M., Lesur O., Kutsiogiannis J., Friedrich J., Klinger J.R., Qushmaq I., Burry L., Khwaja K., Sheppard J.-A., Warkentin T.E., Tkaczyk A., Clarke F., Hall R., Rocker G., Julien L., Wright D., Roy C., Theriault J., Pleasance S., Meade M., Hand L., Freitag A., Wynne C., Duffett M., Kho M., Granton J., Matte A., Farias P., Chu L., Brockest N., Go S., McGrath-Chong M., Dennis M., Lipkus M., Stern E., Albert R., Langevin S., Lauzier F., Turgeon A.F., Blais M., Beauparlant M., Asselin J., Gagne C., Thibodeau M., Poirier G., Neas I., Spearson S., Watpool I., McArdle T., Gaudert C., Marchand P., Davidson C., Pagliarello J., Lewis M.-J., Gosselin A.-A., Deroy P., Ethier C., Tirgari S., Steinberg L., McDonald R., Sivanantham V., Bandayrel K., Quittnat-Pelletier F., Kramer-Kile M., Brown M., Kim S., Fowler R., Marinoff N., Code K., Bojilov B., Parsotam D., Marshall J., Smith O., Fry B., Porretta K., Lee Y., Morrissey J., Wen V., Fleury S., Godfrey N., Hammond S., Mann E., Myers M., Robinson A., Keenan S., Reynolds S., Svetik M., Osch M.V., Chittock D., Dhingra V., Gardner M., Logie S., Foster D., Autio R., Davies D., Ganz P., Smith L., Ashley B.J., Mans S., Doig C., Knox L., Wilson C., Champagne K., Ferguson N., Stevenson J., Elman J., Kutsogiannis J., Thompson P., Whalen N., Lellouche F., Ferland M.-C., Dussault P., Jacob C., Morneau M.-E., Laberge N., Karachi T., Irwin M., Chan C., Sonnema L., Marsh K., Maurer J., Kreidl T., Varden C., Kinjerski C., Banici L., Havell L., Wood G., Auld F., Atkins L., Proulx S., Hollinger G., Shende V., Belcastro V., Plaxton B., Foss A., Paunovic B., Wiebe K., Marten N., Eisenstat J., Doerle T., Sharpe M., Madady M., Cooper J., Davies A., Weatherburn C., Board J., Bennett V., Ramakrishnan N., Bird S., Potter J., O'Connor A., Ankers S., Cade J., Hebert P., Murphy E., Foxall J., Skrobik Y., Harvey J., Chitu S., Sirois C., Nadon C., Dolle S., Barge D., Caf T., Howe B., Bellomo R., Eastwood G., Peck L., Goldsmith D., O'Sullivan K., Ernest D., Radford S., Whitfield A., Cross A., Eliott S., Sidhu J., Mercer I., Hamilton A., Botha J., Vuat J., Allsop S., Fowler N., Crozier T., Barrett J., Wright C., Galt P., Culhane C., Ioannidis R., Burton S., Reily M., Weeraratna C., Seppelt I., Weisbrodt L., Bond R., Evans D., Rivett J., O'Connor S., Poole A., Woolfe C., Rajbhandari D., Rees C., Edington J., Fletcher J., Smith J., Boschert C., Reece G., Sara T., Nand K., Bersten A., Gallus A., Matheson E., O'Callaghan M., Orford N., Elderkin T., Fraser M., Bone A., Salerno T., Kinmonth A., Arora S., O'Bree B., Shepherd K., Davey-Quinn A., Sterba M., Johnson B.R., Xu R., Hill F., Ramadoss R., Wood J., Crowther M., Cook D., Guyatt G., Zytaruk N., McDonald E., Williamson D., Albert M., Dodek P., Finfer S., Vallance S., Heels-Ansdell D., McIntyre L., Mehta S., Lamontagne F., Muscedere J., Jacka M., Lesur O., Kutsiogiannis J., Friedrich J., Klinger J.R., Qushmaq I., Burry L., Khwaja K., Sheppard J.-A., Warkentin T.E., Tkaczyk A., Clarke F., Hall R., Rocker G., Julien L., Wright D., Roy C., Theriault J., Pleasance S., Meade M., Hand L., Freitag A., Wynne C., Duffett M., Kho M., Granton J., Matte A., Farias P., Chu L., Brockest N., Go S., McGrath-Chong M., Dennis M., Lipkus M., Stern E., Albert R., Langevin S., Lauzier F., Turgeon A.F., Blais M., Beauparlant M., Asselin J., Gagne C., Thibodeau M., Poirier G., Neas I., Spearson S., Watpool I., McArdle T., Gaudert C., Marchand P., Davidson C., Pagliarello J., Lewis M.-J., Gosselin A.-A., Deroy P., Ethier C., Tirgari S., Steinberg L., McDonald R., Sivanantham V., Bandayrel K., Quittnat-Pelletier F., Kramer-Kile M., Brown M., Kim S., Fowler R., Marinoff N., Code K., Bojilov B., Parsotam D., Marshall J., Smith O., Fry B., Porretta K., Lee Y., Morrissey J., Wen V., Fleury S., Godfrey N., Hammond S., Mann E., Myers M., Robinson A., Keenan S., Reynolds S., Svetik M., Osch M.V., Chittock D., Dhingra V., Gardner M., Logie S., Foster D., Autio R., Davies D., Ganz P., Smith L., Ashley B.J., Mans S., Doig C., Knox L., Wilson C., Champagne K., Ferguson N., Stevenson J., Elman J., Kutsogiannis J., Thompson P., Whalen N., Lellouche F., Ferland M.-C., Dussault P., Jacob C., Morneau M.-E., Laberge N., Karachi T., Irwin M., Chan C., Sonnema L., Marsh K., Maurer J., Kreidl T., Varden C., Kinjerski C., Banici L., Havell L., Wood G., Auld F., Atkins L., Proulx S., Hollinger G., Shende V., Belcastro V., Plaxton B., Foss A., Paunovic B., Wiebe K., Marten N., Eisenstat J., Doerle T., Sharpe M., Madady M., Cooper J., Davies A., Weatherburn C., Board J., Bennett V., Ramakrishnan N., Bird S., Potter J., O'Connor A., Ankers S., and Cade J.

Abstract

Background: Thrombocytopenia occurs in 20% to 45% of critically ill medical-surgical patients. The 4Ts heparin-induced thrombocytopenia (HIT) score (with 4 domains: Thrombocytopenia, Timing of thrombocytopenia, Thrombosis and o. Ther reason[s] for thrombocytopenia) might reliably identify patients at low risk for HIT. Interobserver agreement on 4Ts scoring is uncertain in this setting. Objective(s): To evaluate whether a published clinical prediction rule (the "4Ts score") reliably rules out HIT in "low-risk" intensive care unit (ICU) patients as assessed by research coordinators (who prospectively scored) and 2 adjudicators (who scored retrospectively) during an international heparin thromboprophylaxis trial (PROTECT, NCT00182143). Method(s): Of 3746 medical-surgical ICU patients in PROTECT, 794 met the enrollment criteria for this HIT substudy. Enrollment was predicated on one of the following occurring in ICU: platelets less than 50 x 109/L, platelets decreased to 50% of ICU admission value (if admission value <100 x 109/L), any venous thrombosis, or if HIT was otherwise clinically suspected. Independently, 4Ts scores were completed in real time by research coordinators blinded to study drug and laboratory HIT results, and retrospectively by 2 adjudicators blinded to study drug, laboratory HIT results, and research coordinators' scores; the adjudicators arrived at consensus in all cases. Of the 763 patients, 474 had a central or local laboratory HIT test performed and had 4Ts scoring by adjudicators; 432 were scored by trained research coordinators. Heparin-induced thrombocytopenia was defined by a centrally performed positive serotonin release assay (SRA). Result(s): Of the 474 patients with central adjudication, 407 (85.9%) had a 4Ts score of 3 or lower, conferring a low pretest probability (PTP) of HIT; of these, 6 (1.5% [95% confidence interval, 0.7%-3.2%) had a positive SRA. Fifty-nine (12.4%) had a moderate PTP (4Ts score of 4-5); of these, 4 (6.8%) had a p

Published
2014

7. Heparin-induced thrombocytopenia in the critically ill: Interpreting the 4Ts test in a randomized trial.

Author

Hebert P., Murphy E., Foxall J., Skrobik Y., Harvey J., Chitu S., Sirois C., Nadon C., Dolle S., Barge D., Caf T., Howe B., Bellomo R., Eastwood G., Peck L., Goldsmith D., O'Sullivan K., Ernest D., Radford S., Whitfield A., Cross A., Eliott S., Sidhu J., Mercer I., Hamilton A., Botha J., Vuat J., Allsop S., Fowler N., Crozier T., Barrett J., Wright C., Galt P., Culhane C., Ioannidis R., Burton S., Reily M., Weeraratna C., Seppelt I., Weisbrodt L., Bond R., Evans D., Rivett J., O'Connor S., Poole A., Woolfe C., Rajbhandari D., Rees C., Edington J., Fletcher J., Smith J., Boschert C., Reece G., Sara T., Nand K., Bersten A., Gallus A., Matheson E., O'Callaghan M., Orford N., Elderkin T., Fraser M., Bone A., Salerno T., Kinmonth A., Arora S., O'Bree B., Shepherd K., Davey-Quinn A., Sterba M., Johnson B.R., Xu R., Hill F., Ramadoss R., Wood J., Crowther M., Cook D., Guyatt G., Zytaruk N., McDonald E., Williamson D., Albert M., Dodek P., Finfer S., Vallance S., Heels-Ansdell D., McIntyre L., Mehta S., Lamontagne F., Muscedere J., Jacka M., Lesur O., Kutsiogiannis J., Friedrich J., Klinger J.R., Qushmaq I., Burry L., Khwaja K., Sheppard J.-A., Warkentin T.E., Tkaczyk A., Clarke F., Hall R., Rocker G., Julien L., Wright D., Roy C., Theriault J., Pleasance S., Meade M., Hand L., Freitag A., Wynne C., Duffett M., Kho M., Granton J., Matte A., Farias P., Chu L., Brockest N., Go S., McGrath-Chong M., Dennis M., Lipkus M., Stern E., Albert R., Langevin S., Lauzier F., Turgeon A.F., Blais M., Beauparlant M., Asselin J., Gagne C., Thibodeau M., Poirier G., Neas I., Spearson S., Watpool I., McArdle T., Gaudert C., Marchand P., Davidson C., Pagliarello J., Lewis M.-J., Gosselin A.-A., Deroy P., Ethier C., Tirgari S., Steinberg L., McDonald R., Sivanantham V., Bandayrel K., Quittnat-Pelletier F., Kramer-Kile M., Brown M., Kim S., Fowler R., Marinoff N., Code K., Bojilov B., Parsotam D., Marshall J., Smith O., Fry B., Porretta K., Lee Y., Morrissey J., Wen V., Fleury S., Godfrey N., Hammond S., Mann E., Myers M., Robinson A., Keenan S., Reynolds S., Svetik M., Osch M.V., Chittock D., Dhingra V., Gardner M., Logie S., Foster D., Autio R., Davies D., Ganz P., Smith L., Ashley B.J., Mans S., Doig C., Knox L., Wilson C., Champagne K., Ferguson N., Stevenson J., Elman J., Kutsogiannis J., Thompson P., Whalen N., Lellouche F., Ferland M.-C., Dussault P., Jacob C., Morneau M.-E., Laberge N., Karachi T., Irwin M., Chan C., Sonnema L., Marsh K., Maurer J., Kreidl T., Varden C., Kinjerski C., Banici L., Havell L., Wood G., Auld F., Atkins L., Proulx S., Hollinger G., Shende V., Belcastro V., Plaxton B., Foss A., Paunovic B., Wiebe K., Marten N., Eisenstat J., Doerle T., Sharpe M., Madady M., Cooper J., Davies A., Weatherburn C., Board J., Bennett V., Ramakrishnan N., Bird S., Potter J., O'Connor A., Ankers S., Cade J., Hebert P., Murphy E., Foxall J., Skrobik Y., Harvey J., Chitu S., Sirois C., Nadon C., Dolle S., Barge D., Caf T., Howe B., Bellomo R., Eastwood G., Peck L., Goldsmith D., O'Sullivan K., Ernest D., Radford S., Whitfield A., Cross A., Eliott S., Sidhu J., Mercer I., Hamilton A., Botha J., Vuat J., Allsop S., Fowler N., Crozier T., Barrett J., Wright C., Galt P., Culhane C., Ioannidis R., Burton S., Reily M., Weeraratna C., Seppelt I., Weisbrodt L., Bond R., Evans D., Rivett J., O'Connor S., Poole A., Woolfe C., Rajbhandari D., Rees C., Edington J., Fletcher J., Smith J., Boschert C., Reece G., Sara T., Nand K., Bersten A., Gallus A., Matheson E., O'Callaghan M., Orford N., Elderkin T., Fraser M., Bone A., Salerno T., Kinmonth A., Arora S., O'Bree B., Shepherd K., Davey-Quinn A., Sterba M., Johnson B.R., Xu R., Hill F., Ramadoss R., Wood J., Crowther M., Cook D., Guyatt G., Zytaruk N., McDonald E., Williamson D., Albert M., Dodek P., Finfer S., Vallance S., Heels-Ansdell D., McIntyre L., Mehta S., Lamontagne F., Muscedere J., Jacka M., Lesur O., Kutsiogiannis J., Friedrich J., Klinger J.R., Qushmaq I., Burry L., Khwaja K., Sheppard J.-A., Warkentin T.E., Tkaczyk A., Clarke F., Hall R., Rocker G., Julien L., Wright D., Roy C., Theriault J., Pleasance S., Meade M., Hand L., Freitag A., Wynne C., Duffett M., Kho M., Granton J., Matte A., Farias P., Chu L., Brockest N., Go S., McGrath-Chong M., Dennis M., Lipkus M., Stern E., Albert R., Langevin S., Lauzier F., Turgeon A.F., Blais M., Beauparlant M., Asselin J., Gagne C., Thibodeau M., Poirier G., Neas I., Spearson S., Watpool I., McArdle T., Gaudert C., Marchand P., Davidson C., Pagliarello J., Lewis M.-J., Gosselin A.-A., Deroy P., Ethier C., Tirgari S., Steinberg L., McDonald R., Sivanantham V., Bandayrel K., Quittnat-Pelletier F., Kramer-Kile M., Brown M., Kim S., Fowler R., Marinoff N., Code K., Bojilov B., Parsotam D., Marshall J., Smith O., Fry B., Porretta K., Lee Y., Morrissey J., Wen V., Fleury S., Godfrey N., Hammond S., Mann E., Myers M., Robinson A., Keenan S., Reynolds S., Svetik M., Osch M.V., Chittock D., Dhingra V., Gardner M., Logie S., Foster D., Autio R., Davies D., Ganz P., Smith L., Ashley B.J., Mans S., Doig C., Knox L., Wilson C., Champagne K., Ferguson N., Stevenson J., Elman J., Kutsogiannis J., Thompson P., Whalen N., Lellouche F., Ferland M.-C., Dussault P., Jacob C., Morneau M.-E., Laberge N., Karachi T., Irwin M., Chan C., Sonnema L., Marsh K., Maurer J., Kreidl T., Varden C., Kinjerski C., Banici L., Havell L., Wood G., Auld F., Atkins L., Proulx S., Hollinger G., Shende V., Belcastro V., Plaxton B., Foss A., Paunovic B., Wiebe K., Marten N., Eisenstat J., Doerle T., Sharpe M., Madady M., Cooper J., Davies A., Weatherburn C., Board J., Bennett V., Ramakrishnan N., Bird S., Potter J., O'Connor A., Ankers S., and Cade J.

Abstract

Background: Thrombocytopenia occurs in 20% to 45% of critically ill medical-surgical patients. The 4Ts heparin-induced thrombocytopenia (HIT) score (with 4 domains: Thrombocytopenia, Timing of thrombocytopenia, Thrombosis and o. Ther reason[s] for thrombocytopenia) might reliably identify patients at low risk for HIT. Interobserver agreement on 4Ts scoring is uncertain in this setting. Objective(s): To evaluate whether a published clinical prediction rule (the "4Ts score") reliably rules out HIT in "low-risk" intensive care unit (ICU) patients as assessed by research coordinators (who prospectively scored) and 2 adjudicators (who scored retrospectively) during an international heparin thromboprophylaxis trial (PROTECT, NCT00182143). Method(s): Of 3746 medical-surgical ICU patients in PROTECT, 794 met the enrollment criteria for this HIT substudy. Enrollment was predicated on one of the following occurring in ICU: platelets less than 50 x 109/L, platelets decreased to 50% of ICU admission value (if admission value <100 x 109/L), any venous thrombosis, or if HIT was otherwise clinically suspected. Independently, 4Ts scores were completed in real time by research coordinators blinded to study drug and laboratory HIT results, and retrospectively by 2 adjudicators blinded to study drug, laboratory HIT results, and research coordinators' scores; the adjudicators arrived at consensus in all cases. Of the 763 patients, 474 had a central or local laboratory HIT test performed and had 4Ts scoring by adjudicators; 432 were scored by trained research coordinators. Heparin-induced thrombocytopenia was defined by a centrally performed positive serotonin release assay (SRA). Result(s): Of the 474 patients with central adjudication, 407 (85.9%) had a 4Ts score of 3 or lower, conferring a low pretest probability (PTP) of HIT; of these, 6 (1.5% [95% confidence interval, 0.7%-3.2%) had a positive SRA. Fifty-nine (12.4%) had a moderate PTP (4Ts score of 4-5); of these, 4 (6.8%) had a p

Published
2014

8. Torus knots and the rational DAHA

Author

Gorsky, E, Gorsky, E, Oblomkov, A, Rasmussen, J, Shende, V, Gorsky, E, Gorsky, E, Oblomkov, A, Rasmussen, J, and Shende, V

Abstract

We conjecturally extract the triply graded Khovanov-Rozansky homology of the (m;n) torus knot from the unique finite-dimensional simple representation of the rational DAHA of type A, rank n - 1, and central character m/n. The conjectural differentials of Gukov, Dunfield, and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov-Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to q; t-Catalan numbers and to previous conjectures of the last three authors relating knot homology to Hilbert schemes on singular curves.

Published
2014

9. Large N duality, lagrangian cycles, and algebraic knots

Author

Diaconescu, D. -E., Shende, V., Vafa, C., Diaconescu, D. -E., Shende, V., and Vafa, C.

Abstract

We consider knot invariants in the context of large $N$ transitions of topological strings. In particular we consider aspects of Lagrangian cycles associated to knots in the conifold geometry. We show how these can be explicity constructed in the case of algebraic knots. We use this explicit construction to explain a recent conjecture relating study of stable pairs on algebraic curves with HOMFLY polynomials. Furthermore, for torus knots, using the explicit construction of the Lagrangian cycle, we also give a direct A-model computation and recover the HOMFLY polynomial for this case., Comment: 53 pages, latex

Published
2011
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10. Large N duality, lagrangian cycles, and algebraic knots

Author

Diaconescu, D. -E., Shende, V., Vafa, C., Diaconescu, D. -E., Shende, V., and Vafa, C.

Abstract

We consider knot invariants in the context of large $N$ transitions of topological strings. In particular we consider aspects of Lagrangian cycles associated to knots in the conifold geometry. We show how these can be explicity constructed in the case of algebraic knots. We use this explicit construction to explain a recent conjecture relating study of stable pairs on algebraic curves with HOMFLY polynomials. Furthermore, for torus knots, using the explicit construction of the Lagrangian cycle, we also give a direct A-model computation and recover the HOMFLY polynomial for this case., Comment: 53 pages, latex

Published
2011
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11. A short proof of the G\'ottsche conjecture

Author

Kool, M., Shende, V., Thomas, R. P., Kool, M., Shende, V., and Thomas, R. P.

Abstract

We prove that for a sufficiently ample line bundle $L$ on a surface $S$, the number of $\delta$-nodal curves in a general $\delta$-dimensional linear system is given by a universal polynomial of degree $\delta$ in the four numbers $L^2,\,L.K_S,\,K_S^2$ and $c_2(S)$. The technique is a study of Hilbert schemes of points on curves on a surface, using the BPS calculus of [PT3] and the computation of tautological integrals on Hilbert schemes by Ellingsrud, G\"ottsche and Lehn. We are also able to weaken the ampleness required, from G\"ottsche's $(5\delta-1)$-very ample to $\delta$-very ample., Comment: 8 pages. Published version

Published
2010
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12. A short proof of the G\'ottsche conjecture

Author

Kool, M., Shende, V., Thomas, R. P., Kool, M., Shende, V., and Thomas, R. P.

Abstract

We prove that for a sufficiently ample line bundle $L$ on a surface $S$, the number of $\delta$-nodal curves in a general $\delta$-dimensional linear system is given by a universal polynomial of degree $\delta$ in the four numbers $L^2,\,L.K_S,\,K_S^2$ and $c_2(S)$. The technique is a study of Hilbert schemes of points on curves on a surface, using the BPS calculus of [PT3] and the computation of tautological integrals on Hilbert schemes by Ellingsrud, G\"ottsche and Lehn. We are also able to weaken the ampleness required, from G\"ottsche's $(5\delta-1)$-very ample to $\delta$-very ample., Comment: 8 pages. Published version

Published
2010
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