Khovanov homology of alternating links and SU(2) representations of the link group

Representation spaces of pretzel knots

“…For torus knots of type (2, p) this was observed by Kronheimer and Mrowka [11, Observation 1 .1]. For an arbitrary 2-bridge knot or 2-component link this was proved by Lewallen [14](in the current version by use of an unpublished result of Shumakovitch [19]). More precisely, he shows that Khovanov homology of a one or two component alternating link is isomorphic to the integer homology of R bd (K; i), where R bd (K; i) ⊆ R(K; i) is the subspace of binary dihedral representations.…”

“…We just pick α pq = π/|p| and α qr = π/|q|, both satisfying the required congruences for α pq and α qr in this case. The triangle inequality (14) |r| inside this interval, and the choice of one of them for the number α rp /π will satisfy the congruence r α rp ≡ r π (mod 2π ).…”

“…We choose α pq = 2π/|p| and α qr = π/|q|. The triangle inequality (14) In each situation, it follows from Proposition 4.1 above that the corresponding representation is not binary dihedral.…”

Trace-free $$\mathrm{SL}(2,{\pmb {\mathbb {C}}})$$ SL ( 2 , C ) -representations of arborescent links

“…Researchers have paid attention to such representations for long. For a knot K, Lin [6] defined an invariant h(K) roughly by counting (with signs) conjugacy classes of trace-free SU(2)-representations of K, and showed that h(K) equals half of the signature of K. Interestingly, it was shown in [5] that if L is an alternating link or a 2-component link, then its Khovanov homology is isomorphic to the singular homology of the space of binary dihedral representations (a special kind of trace-free SU(2)-representations) of L, as graded abelian groups. Also interesting is that, by the result of [7], each trace-free SL(2, C)-representation of L gives rise to a representation π 1 (M 2 (L)) → SL(2, C), where M 2 (L) is the double covering of S 3 branched along L.…”