The rational Khovanov homology of 3-strand pretzel links

Abstract

Abstract. The 3-strand pretzel knots and links are a well-studied source of examples in knot theory. However, while there have been computations of the Khovanov homology of some sub-families of 3-strand pretzel knots, no general formula has been given for all of them. We give a general formula for the unreduced Khovanov homology of all 3-strand pretzel links, over the rational numbers.

Rasmussen Invariants of Some 4-Strand Pretzel Knots

“…The s invarinats of most 3-strand pretzel knots were computed by Suzuki [14] and those of all remaining 3-strand pretzel knots and links were computed by Manion [11]. The s invariants of general pretzel knots with only one negatively twisted strand were computed by Kawamura [6], which are turned out to be equal to −σ and 2τ .…”

“…Using his formula it is easy to see that, for any odd integer a ≥ 3, the negative value of the signature of pretzel knot P (−a, a + 2, a + 1) is equal to 2. On the other hand, Manion's formula in [11] tells us that s invariant of P (−a, a + 2, a + 1) is zero for any positive integer a. Thus there is an infinite family of 3-strand pretzel knots whose s invariants are not equal to −σ.…”

“…The rational Khovanov homology of D 0 ∼ = P (−3, 5, 4) can be computed by the work of Manion [11]. Since P (−3, 5, 4) is isotopic to P (−3, 4, 5), we can compute the Khovanov homology of P (−3, 4, 5) from Manion's formula Kh(P (−3, 4, 5)) = q −13 t −7 L −3.4.5 ⊕ q −1 t 0 U −3,4,5 whose nonzero groups are all depicted in Figure 8.…”

“…Manion [11] computed rational Khovanov homologies of all non quasi-alternating 3-strand pretzel knots and links and found the Rasmussen invariants of all 3-strand pretzel knots and links.…”

Rasmussen and Ozsváth–Szabó invariants of a family of general pretzel knots

“…The Rasmussen invariants of 3-strand pretzel knots were partly computed by Suzuki [15]. Then Manion [11] calculated the rational Khovanov homology and the Rasmussen invariants of all 3-strand pretzel knots and links. The Ozsváth-Szabó knot Floer homology τ -invariants have been evaluated for certain families of pretzel knots in [3,4].…”