Hypercube Approach

In hypercube approach with each knot diagram one associates a hypercube of its resolutions. At hypercube vertices stand graded spaces with certain quantum dimensions (which can be negative for virtual knots). Hypercube edges correspond to flips between resolutions at particular vertices of the knot diagram, and to cut-and-join morphisms between spaces at hypercube vertices. If morphisms are commuting, Abelian quiver can be associated with the hypercube. HOMFLY polynomials are made from quantum (q-graded) dimensions of spaces at hypercube. Khovanov-Rozansky and superpolynomials - from q-graded dimensions of cohomologies of the Abelian complex. Explicit construction of commuting morphisms is known for N=2 (Jones polynomials) and is conjectured for arbitrary N.

• Original papers
  • L.Kauffman, Topology 26 (1987) 395-407;
  • L.Kauffman, Trans.Amer.Math.Soc. 311 (1989) 697-710;
  • L.Kauffman and P.Vogel, J.Knot Theory Ramifications 1 (1992) 59-104;
  • M.Khovanov, Experimental Math. 12 (2003) no.3, 365374, math/0201306;
  • M.Khovanov, J.Knot theory and its Ramifications 14 (2005) no.1, 111-130, math/0302060;
  • M.Khovanov, Algebr. Geom. Topol. 4 (2004) 1045-1081, math/0304375;
  • M.Khovanov, Int.J.Math. 18 (2007) no.8, 869885, math/0510265;
  • M.Khovanov, math/0605339;
  • M.Khovanov, arXiv:1008.5084;
  • D.Bar-Natan, Algebraic and Geometric Topology 2 (2002) 337-370, math/0201043;
  • D.Bar-Natan, Geom.Topol. 9 (2005) 1443-1499, math/0410495;
  • D.Bar-Natan, J.Knot Theory Ramifications 16 (2007) no.3, 243255, math/0606318
• Recent Advances
  • V.Dolotin and A.Morozov, JHEP 1301 (2013) 065, arXiv:1208.4994(morphisms as cut-and-join operators);
  • V.Dolotin and A.Morozov, J.Phys.:Conf.Ser. 411 012013, arXiv:1209.5109;
  • V.Dolotin and A.Morozov, Nucl.Phys. B878 (2014) 12-81, arXiv:1308.5759 (resolutions, relevant for arbitrary N);
  • A.Anokhina and A.Morozov, JHEP 07 (2014) 063, arXiv:1403.8087;
  • A.Morozov, And.Morozov and A.Popolitov, Phys.Lett. B (2015), arXiv:1506.07516 (matrix-model formalism and AMM/EO-like recursion for quantum dimensions);
  • A.Morozov, And.Morozov and A.Popolitov, to be published (matrix-model formalism and AMM/EO-like recursion for quantum dimensions)
• Virtual knots and links
  • A.Morozov, And.Morozov and Ant.Morozov, Phys.Lett.B737 (2014) 48-56, arXiv:1407.6319;
  • L.Bishler, A.Morozov, And.Morozov and Ant.Morozov, Int.J.Mod.Phys.A30 (2015) 1550074, arXiv:1411.2569;
  • A.Morozov, And.Morozov and A.Popolitov, Phys.Lett. B (2015), arXiv:1506.07516 (matrix-model formalism and AMM/EO-like recursion for quantum dimensions);
  • A.Morozov, And.Morozov and A.Popolitov, to be published (matrix-model formalism and AMM/EO-like recursion for quantum dimensions)