Splitting the spectral flow and the SU(3) Casson invariant for spliced sums. (English) Zbl 1180.57019
The aim of the paper under review is to show that the SU(3) Casson invariant defined by Boden, Herald and Kirk for spliced sums along \((2, q_1)\) and \((2, q_2)\) torus knots is 16 times the product of the SU(2) Casson knot invariants of the \((2, q_1)\) and \((2, q_2)\) torus knots.
Note that the spliced sum of integral homology 3-spheres \(M_1\) and \(M_2\) along knots \(K_1\subset M_1\) and \(K_2 \subset M_2\) is the integral homology 3-sphere obtained by gluing \(\partial (M_1 \setminus \overline{N(K_1)})\) and \(\partial (M_2 \setminus \overline{N(K_2)})\) by identifying the meridian of one knot with the longitude of the other knot, denoted by \(M_1(K_1)\#_T M_2(K_2)\). The SU(2) Casson invariant of integral homology 3-spheres is additive under connected sum, and also additive under spliced sum. Although the SU(3) Casson invariant \(\tau_{\text{SU}(3)}\) is not additive under connected sum, the difference \(\tau_{\text{SU}(3)} - 2 \lambda^2_{\text{SU}(2)}\) is additive under connected sum but not additive under spliced sum.
The paper under review analyzes the SU(3) Casson invariant under spliced sum along torus knots \((2, q_1)\) and \((2, q_2)\). Since the detailed analysis of SU(3) representation varieties of the torus knot complements with computations of the \(\text{ad\, SU}(3)\) spectral flows of the odd signature operator coupled to a path of SU(3) connections can be achieved, and all spectral flows are even, the SU(3) Casson invariant can be computed by identification of irreducible SU(3) representations under spliced sum. In Section 2, the authors give the splitting formula with Lagrangian boundaries for the spectral flow in Theorem 2.10. Section 3 is devoted to the study of the SU(3) representation variety of a spliced sum: an SU(3) representation \(\alpha\) on \(M_1(K_1)\#_T M_2(K_2)\) with \(\alpha|_{M_i \setminus N(K_i)}\) abelian must be trivial via the splicing construction; an SU(3) representation \(\alpha\) with isotropy group of \(\alpha|_{\partial (M_i \setminus N(K_i))}\) not the maximal torus \(T_{\text{SU}(3)}\) has both irreducible restrictions on \(M_i \setminus N(K_i)\) (\(i=1, 2\)) (and this component is a subset of irreducible components and diffeomorphic to \(C = S(U(2) \times U(1))/\mathbb Z_3\) with \(\chi (C) = 0\) as proved in Proposition 3.4); an SU(3) representation \(\alpha\) with isotropy group of \(\alpha|_{\partial (M_i \setminus N(K_i))}\) the maximal torus has three possibilities ((a) both \(\alpha|_{\partial (M_i \setminus N(K_i))}\) irreducible with components diffeomorphic to \(T_{\text{SU}(3)}/\mathbb Z_3\), (b) one of \(\alpha|_{\partial (M_i \setminus N(K_i))}\) irreducible and the other reducible and nonabelian with components diffeomorphic to \(T_{\text{SU}(3)}/U(1)\) (both (a) and (b) cases have zero Euler characteristic number), (c) both \(\alpha|_{\partial (M_i \setminus N(K_i))}\) are reducible and nonabelian with components either \(S^1\) or isolated points) as described in Theorem 3.6.
The key point is that there is no need to count positive dimensional components since all satisfy \(\chi (C_j) = 0\), and the counting will focus on 0-dimensional or isolated components. This reduces the study to the reducible representation stratum on \(M_i \setminus N(K_i)\) for the torus \((2, q)\) knot complement via the method Klassen studied earlier, and the number of SU(3) isolated conjugacy classes of \(M_1(K_1)\#_T M_2(K_2)\) is \(\frac{1}{4} (q_1^2-1)(q_2^2-1)\) (section 5). Section 6 is devoted to the cohomology calculations for torus knots and determines that the spectral flow is even for all the isolated conjugated classes. This gives the positive contribution to the SU(3) Casson invariant with possible multiplicities.
The proof of \(\lambda_{\text{SU}(3)}(M_1(K_1)\#_T M_2(K_2)) = 16 \lambda'_{\text{SU}(2)}(K_1) \cdot \lambda'_{\text{SU}(2)}(K_2)\) does not reveal the reason for the identification between the SU(3) Casson invariant and the product of SU(2) Casson invariant of knots. It makes it hard to predict whether the same relation even holds for other torus knots. It would also be interesting to make sure that the multiplicity for each isolated conjugacy class equals one.
Note that the spliced sum of integral homology 3-spheres \(M_1\) and \(M_2\) along knots \(K_1\subset M_1\) and \(K_2 \subset M_2\) is the integral homology 3-sphere obtained by gluing \(\partial (M_1 \setminus \overline{N(K_1)})\) and \(\partial (M_2 \setminus \overline{N(K_2)})\) by identifying the meridian of one knot with the longitude of the other knot, denoted by \(M_1(K_1)\#_T M_2(K_2)\). The SU(2) Casson invariant of integral homology 3-spheres is additive under connected sum, and also additive under spliced sum. Although the SU(3) Casson invariant \(\tau_{\text{SU}(3)}\) is not additive under connected sum, the difference \(\tau_{\text{SU}(3)} - 2 \lambda^2_{\text{SU}(2)}\) is additive under connected sum but not additive under spliced sum.
The paper under review analyzes the SU(3) Casson invariant under spliced sum along torus knots \((2, q_1)\) and \((2, q_2)\). Since the detailed analysis of SU(3) representation varieties of the torus knot complements with computations of the \(\text{ad\, SU}(3)\) spectral flows of the odd signature operator coupled to a path of SU(3) connections can be achieved, and all spectral flows are even, the SU(3) Casson invariant can be computed by identification of irreducible SU(3) representations under spliced sum. In Section 2, the authors give the splitting formula with Lagrangian boundaries for the spectral flow in Theorem 2.10. Section 3 is devoted to the study of the SU(3) representation variety of a spliced sum: an SU(3) representation \(\alpha\) on \(M_1(K_1)\#_T M_2(K_2)\) with \(\alpha|_{M_i \setminus N(K_i)}\) abelian must be trivial via the splicing construction; an SU(3) representation \(\alpha\) with isotropy group of \(\alpha|_{\partial (M_i \setminus N(K_i))}\) not the maximal torus \(T_{\text{SU}(3)}\) has both irreducible restrictions on \(M_i \setminus N(K_i)\) (\(i=1, 2\)) (and this component is a subset of irreducible components and diffeomorphic to \(C = S(U(2) \times U(1))/\mathbb Z_3\) with \(\chi (C) = 0\) as proved in Proposition 3.4); an SU(3) representation \(\alpha\) with isotropy group of \(\alpha|_{\partial (M_i \setminus N(K_i))}\) the maximal torus has three possibilities ((a) both \(\alpha|_{\partial (M_i \setminus N(K_i))}\) irreducible with components diffeomorphic to \(T_{\text{SU}(3)}/\mathbb Z_3\), (b) one of \(\alpha|_{\partial (M_i \setminus N(K_i))}\) irreducible and the other reducible and nonabelian with components diffeomorphic to \(T_{\text{SU}(3)}/U(1)\) (both (a) and (b) cases have zero Euler characteristic number), (c) both \(\alpha|_{\partial (M_i \setminus N(K_i))}\) are reducible and nonabelian with components either \(S^1\) or isolated points) as described in Theorem 3.6.
The key point is that there is no need to count positive dimensional components since all satisfy \(\chi (C_j) = 0\), and the counting will focus on 0-dimensional or isolated components. This reduces the study to the reducible representation stratum on \(M_i \setminus N(K_i)\) for the torus \((2, q)\) knot complement via the method Klassen studied earlier, and the number of SU(3) isolated conjugacy classes of \(M_1(K_1)\#_T M_2(K_2)\) is \(\frac{1}{4} (q_1^2-1)(q_2^2-1)\) (section 5). Section 6 is devoted to the cohomology calculations for torus knots and determines that the spectral flow is even for all the isolated conjugated classes. This gives the positive contribution to the SU(3) Casson invariant with possible multiplicities.
The proof of \(\lambda_{\text{SU}(3)}(M_1(K_1)\#_T M_2(K_2)) = 16 \lambda'_{\text{SU}(2)}(K_1) \cdot \lambda'_{\text{SU}(2)}(K_2)\) does not reveal the reason for the identification between the SU(3) Casson invariant and the product of SU(2) Casson invariant of knots. It makes it hard to predict whether the same relation even holds for other torus knots. It would also be interesting to make sure that the multiplicity for each isolated conjugacy class equals one.
Reviewer: Weiping Li (Stillwater)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 58J30 | Spectral flows |
| 57R57 | Applications of global analysis to structures on manifolds |
Software:
SageMathReferences:
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