A knot invariant via representation spaces

Gauge theory and knot homologies

“…This is precisely the definition of the knot invariant which was introduced and studied in [39,40,41] (see also [42,43]). This invariant, sometimes called Casson-Lin invariant, is well-defined away from the roots of the Alexander polynomial of K and turns out to be equal to the linear combination of more familiar invariants, α ∈ [0, 1], λ α (Y ; K) was constructed in [44] (see also [45,46]) and, therefore, is expected to be the same as (3.8).…”

“…In particular, for Y = S 3 and α = 1 2 we get the original Lin's invariant [39] and the corresponding homology theory categorifying λ 1 2 (Y ; K) was constructed -as symplectic Floer homology (2.19) of the braid representative of K -in [47].…”

Deforming abelian SU(2)-representations of knot groups

“…The proof of theorem 1.1 makes use of a generalization of a result of X.-S. Lin (see [Lin92]): let G be a knot group and let m ∈ G be a meridian. A representation ρ: G → SU(2) is called trace-free if tr ρ(m) = 0.…”

Representation spaces of pretzel knots

“…In the case that all irreducible representations are nondegenerate, and so these are isolated points in .KI i/, the number h.K/ is just a signed count of these conjugacy classes of irreducible representations. Surprisingly, this is related to the signature of K in the following way [15]:…”

“…Lin [15] has defined a knot invariant, that he denotes h.K/, from the representation space .KI i/ D R.KI i/= SU.2/ considered here. In the case that all irreducible representations are nondegenerate, and so these are isolated points in .KI i/, the number h.K/ is just a signed count of these conjugacy classes of irreducible representations.…”