On some p-differential graded link homologies

Abstract We show that the triply graded Khovanov–Rozansky homology of knots and links over a field of positive odd characteristic p descends to an invariant in the homotopy category finite-dimensional p-complexes. A p-extended differential on the triply graded homology discovered by Cautis is compatible with the p-DG structure. As a consequence, we get a categorification of the Jones polynomial evaluated at a $2p$th root of unity.

[1]  E. Wagner,et al.  Symmetric Khovanov-Rozansky link homologies , 2018, Journal de l’École polytechnique — Mathématiques.

[2]  R. Sarpong,et al.  Bio-inspired synthesis of xishacorenes A, B, and C, and a new congener from fuscol† †Electronic supplementary information (ESI) available. See DOI: 10.1039/c9sc02572c , 2019, Chemical science.

[3]  David E. V. Rose,et al.  Annular Evaluation and Link Homology , 2018, 1802.04131.

[4]  B. Webster,et al.  Knot Invariants and Higher Representation Theory , 2013, 1309.3796.

[5]  Sabin Cautis.  Remarks on coloured triply graded link invariants , 2016, 1611.09924.

[6]  You Qi,et al.  A categorification of the Burau representation at prime roots of unity , 2013, 1312.7692.

[7]  You Qi,et al.  Categorification at prime roots of unity and hopfological finiteness , 2015, 1509.00438.

[8]  C. Ricketts.  Pages on Art , 2015 .

[9]  M. Khovanov,et al.  An approach to categorification of some small quantum groups II , 2012, 1208.0616.

[10]  M. Khovanov,et al.  Positive half of the Witt algebra acts on triply graded link homology , 2013, 1305.1642.

[11]  You Qi.  Hopfological algebra , 2012, Compositio Mathematica.

[12]  R. Rouquier.  Khovanov-Rozansky homology and 2-braid groups , 2012, 1203.5065.

[13]  Mikhail Khovanov,et al.  Diagrammatics for Soergel Categories , 2009, Int. J. Math. Math. Sci..

[14]  M. Khovanov,et al.  Matrix factorizations and link homology II , 2005, math/0505056.

[15]  M. Khovanov,et al.  Matrix factorizations and link homology , 2004, math/0401268.

[16]  Mikhail Khovanov,et al.  Triply-graded link homology and Hochschild homology of Soergel bimodules , 2005, math/0510265.

[17]  M. Khovanov.  Hopfological algebra and categorification at a root of unity: The first steps , 2005, math/0509083.

[18]  R. Rouquier.  Categorification of the braid groups , 2004, math/0409593.

[19]  M. Khovanov.  A categorification of the Jones polynomial , 1999, math/9908171.

[20]  C. Kassel,et al.  Algèbre Homologique des N-Complexes et Homologie de Hochschild aux Racines de l'Unité , 1998 .

[21]  C. Kassel,et al.  Algèbre Homologique des N-Complexes et Homologie de Hochschild aux Racines de l'Unité , 1997, q-alg/9705001.

[22]  L. Crane,et al.  Four‐dimensional topological quantum field theory, Hopf categories, and the canonical bases , 1994, hep-th/9405183.

[23]  V. Turaev,et al.  Ribbon graphs and their invaraints derived from quantum groups , 1990 .

[24]  Edward Witten,et al.  Quantum field theory and the Jones polynomial , 1989 .

[25]  E. Spanier.  The Mayer homology theory , 1949 .

[26]  W. Mayer.  A New Homology Theory. II , 1942 .

[27]  W. Mayer.  A New Homology Theory , 1942 .