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We prove that if $D$ is a diagram of a classical link $L$ and $0=\\phi_0,\\phi_1,\\dots,\\phi_{n-1}$ are the invariant factors of an adjusted Goeritz matrix of $D$, then the group $\\mathcal{D}_{A}(D)$ of Dehn colorings of $D$ with values in $A$ is isomorphic to the direct product of $A$ and $A=A(\\phi_{0}),A(\\phi_1),\\dots,A(\\phi_{n-1})$. It follows that the Dehn coloring groups of $L$ are isomorphic to those of a connected sum of torus links $T_{(2,\\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1804.02700","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-04-08T14:46:02Z","cross_cats_sorted":[],"title_canon_sha256":"5a8017103935bd0728b170eb5332d64fb00fb5c4c29b36e34ba043a9fe19c580","abstract_canon_sha256":"264d9305752303240733a17c4abfa73dc7c7961cd0126c01c694e4566f691d27"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:30.500093Z","signature_b64":"f8EpECuqyxMUOM02qfxCfTbRczdcuKGBXo2EtFe6MtTt4T+QB6pKOhItjXnuSHj39kmpRAPNkY0I1L2WVG2dDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28bc47c8b20fbfc10ec1dd56797d655ba06044c565df91995e97e5272c8e7d13","last_reissued_at":"2026-05-18T00:00:30.499520Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:30.499520Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on Dehn colorings and invariant factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Derek A. Smith, Lorenzo Traldi, William Watkins","submitted_at":"2018-04-08T14:46:02Z","abstract_excerpt":"If $A$ is an abelian group and $\\phi$ is an integer, let $A(\\phi)$ be the subgroup of $A$ consisting of elements $a \\in A$ such that $\\phi \\cdot a=0$. We prove that if $D$ is a diagram of a classical link $L$ and $0=\\phi_0,\\phi_1,\\dots,\\phi_{n-1}$ are the invariant factors of an adjusted Goeritz matrix of $D$, then the group $\\mathcal{D}_{A}(D)$ of Dehn colorings of $D$ with values in $A$ is isomorphic to the direct product of $A$ and $A=A(\\phi_{0}),A(\\phi_1),\\dots,A(\\phi_{n-1})$. 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