ScalingStacks

7.4 Cohomology of the disjoint union and connected sum of knots

Pick diagrams D1,D2D_{1},D_{2} of oriented links L1,L2L_{1},L_{2} and consider a diagram D1βŠ”D2D_{1}\sqcup D_{2} of the disjoint union L1βŠ”L2.L_{1}\sqcup L_{2}. We then have an isomorphism of cochain complexes

π’žβ‘(D1βŠ”D2)=π’žβ‘(D1)βŠ—π’žβ‘(D2){\cal C}(D_{1}\sqcup D_{2})={\cal C}(D_{1})\otimes{\cal C}(D_{2}) (167)

of free graded abelian groups. From the KΓΌnneth formula we derive

0PMA

Proposition 33 There is a short split exact sequence of cohomology groups

0\displaystyle 0 β†’\displaystyle\to βŠ•i,jβˆˆβ„€(β„‹i,j​(D1)βŠ—β„‹kβˆ’i,mβˆ’j​(D2))β†’β„‹k,m​(D1βŠ”D2)β†’\displaystyle{\mathop{\oplus}\limits_{i,j\in\mathbb{Z}}}({\cal H}^{i,j}(D_{1})\otimes{\cal H}^{k-i,m-j}(D_{2}))\to{\cal H}^{k,m}(D_{1}\sqcup D_{2})\to
β†’\displaystyle\to βŠ•i,jβˆˆβ„€Tor1℀​(β„‹i,j​(D1),β„‹kβˆ’i+1,mβˆ’j​(D2))β†’0\displaystyle{\mathop{\oplus}\limits_{i,j\in\mathbb{Z}}}{\mathrm{Tor}}_{1}^{\mathbb{Z}}({\cal H}^{i,j}(D_{1}),{\cal H}^{k-i+1,m-j}(D_{2}))\to 0
0PMB

Corollary 12 For each k,mβˆˆβ„€k,m\in\mathbb{Z} there is an equality of isomorphism classes of abelian groups

β„‹k,m​(L1βŠ”L2)=\displaystyle{\cal H}^{k,m}(L_{1}\sqcup L_{2})= βŠ•i,jβˆˆβ„€(β„‹i,j​(L1)βŠ—β„‹kβˆ’i,mβˆ’j​(L2))βŠ•\displaystyle{\mathop{\oplus}\limits_{i,j\in\mathbb{Z}}}({\cal H}^{i,j}(L_{1})\otimes{\cal H}^{k-i,m-j}(L_{2}))\oplus
βŠ•i,jβˆˆβ„€Tor1℀​(β„‹i,j​(L1),β„‹kβˆ’i+1,mβˆ’j​(L2))\displaystyle{\mathop{\oplus}\limits_{i,j\in\mathbb{Z}}}{\mathrm{Tor}}_{1}^{\mathbb{Z}}({\cal H}^{i,j}(L_{1}),{\cal H}^{k-i+1,m-j}(L_{2}))

Let D1,D2D_{1},D_{2} be diagrams of oriented knots K1,K2.K_{1},K_{2}. We picture D1D_{1} and D2D_{2} schematically as

[Uncaptioned image]

Consider diagrams D3,D4D_{3},D_{4} and D5,D_{5}, depicted below, of oriented links K1βŠ”K2,K_{1}\sqcup K_{2}, K1​#​K2,K_{1}\#K_{2}, and K1​#​(βˆ’K2)K_{1}\#(-K_{2}):

[Uncaptioned image]
[Uncaptioned image]

By resolving the central double point of D5,D_{5}, we get a short exact sequence of complexes of graded abelian groups

0βŸΆπ’žΒ―β€‹(D3)​[βˆ’1]​{βˆ’1}βŸΆπ’žΒ―β€‹(D5)βŸΆπ’žΒ―β€‹(D4)⟢00\longrightarrow\overline{{\cal C}}(D_{3})[-1]\{-1\}\longrightarrow\overline{{\cal C}}(D_{5})\longrightarrow\overline{{\cal C}}(D_{4})\longrightarrow 0 (168)

After shifts, we obtain an exact sequence

0βŸΆπ’žβ‘(D3)​[βˆ’1]​{βˆ’1}βŸΆπ’žβ‘(D5)​[βˆ’1]​{βˆ’2}βŸΆπ’žβ‘(D4)⟢00\longrightarrow{\cal C}(D_{3})[-1]\{-1\}\longrightarrow{\cal C}(D_{5})[-1]\{-2\}\longrightarrow{\cal C}(D_{4})\longrightarrow 0 (169)

which induces, for every integer j,j, a long exact sequence of cohomology groups

β‹―β†’β„‹iβˆ’1,jβˆ’1​(D3)β†’β„‹iβˆ’1,jβˆ’2​(D5)β†’β„‹i,j​(D4)β†’β†’β„‹i,jβˆ’1​(D3)β†’β„‹i,jβˆ’2​(D5)β†’β„‹i+1,j​(D4)β†’β‹―\begin{array}[]{ccccccccc}\cdots&\to&{\cal H}^{i-1,j-1}(D_{3})&\to&{\cal H}^{i-1,j-2}(D_{5})&\to&{\cal H}^{i,j}(D_{4})&\to&\\ &\to&{\cal H}^{i,j-1}(D_{3})&\to&{\cal H}^{i,j-2}(D_{5})&\to&{\cal H}^{i+1,j}(D_{4})&\to&\cdots\end{array} (170)

Since the diagrams D3,D4D_{3},D_{4} and D5D_{5} represent oriented links K1βŠ”K2,K1​#​K2K_{1}\sqcup K_{2},K_{1}\#K_{2} and K1​#​(βˆ’K2),K_{1}\#(-K_{2}), respectively, then in view of PropositionΒ 29, we obtain

0PMC

Proposition 34 For oriented knots K1,K2,K_{1},K_{2}, the isomorphism classes of the abelian groups β„‹i,j​(K1βŠ”K2),β„‹i,j​(K1​#​K2){\cal H}^{i,j}(K_{1}\sqcup K_{2}),{\cal H}^{i,j}(K_{1}\#K_{2}) can be arranged into long exact sequences

β†’β„‹iβˆ’1,jβˆ’1​(K1βŠ”K2)β†’β„‹iβˆ’1,jβˆ’2​(K1​#​K2)β†’β„‹i,j​(K1​#​K2)β†’β†’β„‹i,jβˆ’1​(K1βŠ”K2)β†’β„‹i,jβˆ’2​(K1​#​K2)β†’β„‹i+1,j​(K1​#​K2)β†’\begin{array}[]{ccccccc}\to&{\cal H}^{i-1,j-1}(K_{1}\sqcup K_{2})&\to&{\cal H}^{i-1,j-2}(K_{1}\#K_{2})&\to&{\cal H}^{i,j}(K_{1}\#K_{2})&\to\\ \to&{\cal H}^{i,j-1}(K_{1}\sqcup K_{2})&\to&{\cal H}^{i,j-2}(K_{1}\#K_{2})&\to&{\cal H}^{i+1,j}(K_{1}\#K_{2})&\to\end{array} (171)

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

Mikhail Khovanov

Original source: arXiv:math/9908171v2