Matrix factorizations and link homology. (English) Zbl 1145.57009
The authors construct, for any link \(L\), and each \(n>0\), a doubly graded homology theory \(H_{n}(L)\) of \(L\).
The Euler characteristic \(P_{n}(L)\) of \(H_{n}(L)\) turns out to be a one variable specialization of the HOMFLYPT polynomial, which can be recovered from the representation theory of quantum \(sl(n)\).
For \(n=1\), \(H_{1}(L) \cong\mathbb Z\). For \(n=2\), \(H_{2}(L)\) was defined by the first author in [Duke Math. J. 101, 359–426 (1999; Zbl 0960.57005)] and \(P_{2}(L)\) is the Jones polynomial. For \(n=3, H_{3}(L)\) was defined by the first author in [Algebr. Geom. Topol. 4, 1045–1081 (2004; Zbl 1159.57300)].
The Euler characteristic \(P_{n}(L)\) of \(H_{n}(L)\) turns out to be a one variable specialization of the HOMFLYPT polynomial, which can be recovered from the representation theory of quantum \(sl(n)\).
For \(n=1\), \(H_{1}(L) \cong\mathbb Z\). For \(n=2\), \(H_{2}(L)\) was defined by the first author in [Duke Math. J. 101, 359–426 (1999; Zbl 0960.57005)] and \(P_{2}(L)\) is the Jones polynomial. For \(n=3, H_{3}(L)\) was defined by the first author in [Algebr. Geom. Topol. 4, 1045–1081 (2004; Zbl 1159.57300)].
Reviewer: Lee P. Neuwirth (Princeton)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
