Linking numbers and the elementary ideals of links

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Let L = K 1 ∪ ⋯ ∪ K μ ⊆ S 3 L = {K_1} \cup \, \cdots \cup {K_\mu } \subseteq {S^3} be a tame link of μ ⩾ 2 \mu \geqslant 2 components, and H H the abelianization of G = π 1 ( S 3 − L ) G = {\pi _1}({S^3} - L) . Let L = ( L i j ) \mathcal {L} = ({\mathcal {L}_{ij}}) be the μ × μ \mu \times \mu matrix with entries in Z H \mathbf {Z}H given by L i i = ∑ k ≠ i l ( K i , K k ) ⋅ ( t k − 1 ) \mathcal {L}{_{ii}} = \sum \nolimits _{k \ne i} {l({K_i},{K_k}) \cdot ({t_k} - 1)} and for i ≠ j L i j = l ( K i , K j ) ⋅ ( 1 − t i ) i \ne j\,{\mathcal {L}_{ij}} = l({K_i},{K_j}) \cdot (1 - {t_i}) . Then if 0 > k > μ 0 > k > \mu \[ ∑ i = 0 k − 1 E μ − k + i ( L ) ⋅ ( I H ) 2 i + ( I H ) 2 k = ∑ i = 0 k − 1 E μ − k + i ( L ) ⋅ ( I H ) 2 i + ( I H ) 2 k \sum \limits _{i = 0}^{k - 1} {{E_{\mu - k + i}}(L) \cdot {{(IH)}^{2i}} + {{(IH)}^{2k}} = \sum \limits _{i = 0}^{k - 1} {{E_{\mu - k + i}}(\mathcal {L}) \cdot {{(IH)}^{2i}} + {{(IH)}^{2k}}} } \] Various consequences of this equality are derived, including its application to the reduced elementary ideals. These results are used to give several different characterizations of links in which all the linking numbers are zero.