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This gives counterexamples for the Montesinos-Nakanishi $3$-move conjecture. In general, it is hard to distinguish $p$th Burnside groups of a given link and a trivial link. We give a necessary condition for which $p$th Burnside groups are isomorphic to those of trivial links. The necessary condition gi"},"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":"1801.09863","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-01-30T06:53:45Z","cross_cats_sorted":[],"title_canon_sha256":"dff248a58e10067e81648b6a87059c0d350fdea8a32fc79d2e203dbb7c1c8177","abstract_canon_sha256":"d23b54141dad9b90dfd8137d7fc8b0cc4561ca0eddeafbfae946b46cdaa6f8de"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:46.569675Z","signature_b64":"9PHYSExfWk94cCvPYiCq3JLKzgKAh/zZM9EarZhLCADf4eRvmfeslnhyL0/7sYTF/9XtbK1e8YJy5tbfxztODw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"07e1c226cb1186675bba05c05d3cbb4357f9b447e1397fb9bc13c5180dbbae6b","last_reissued_at":"2026-05-18T00:24:46.569027Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:46.569027Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Burnside groups and $n$-moves for links","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Akira Yasuhara, Haruko A. Miyazawa, Kodai Wada","submitted_at":"2018-01-30T06:53:45Z","abstract_excerpt":"Let $n$ be a positive integer. M. K. Dabkowski and J. H. Przytycki introduced the $n$th Burnside group of links which is preserved by $n$-moves, and proved that for any odd prime $p$ there exist links which are not equivalent to trivial links up to $p$-moves by using their $p$th Burnside groups. This gives counterexamples for the Montesinos-Nakanishi $3$-move conjecture. In general, it is hard to distinguish $p$th Burnside groups of a given link and a trivial link. We give a necessary condition for which $p$th Burnside groups are isomorphic to those of trivial links. 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