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We demonstrate that, for $N \\geq 1$, $\\mathcal{H}_N$ is a $\\mathbb{Z}_2\\oplus\\mathbb{Z}^{\\oplus 3}$-graded $\\mathbb{Q}[a]$-module that is invariant under transverse Markov moves, but not under negative stabilization/de-stabilization. Thus, for $N\\geq 1$, this homology is an invariant for transverse links in the standard contact $S^3$, but not for smoo"},"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":"1308.3152","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-08-14T15:31:29Z","cross_cats_sorted":["math.SG"],"title_canon_sha256":"ef41267fda07bbcd42c101dd082cdc19b473ef60d1e901a22c1f6bc35affee21","abstract_canon_sha256":"e8c35dd3716e28880deb6af77a2a4d4660359a27b39dff6ce5a1ea37437491db"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:31.053165Z","signature_b64":"tfc+2nZ0DyzkJjiJ6sV1dW8JyiF6FT7VVYA2G0QNAjTN6aWNW3qT9T0V9878vo3xjbFXnqdKlnJ6+mYzMRR/BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3fb02bec8d892921503c32fb892ded50d920ec9aa17c373afff51c252b436a25","last_reissued_at":"2026-05-18T01:19:31.052428Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:31.052428Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Family of Transverse Link Homologies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.GT","authors_text":"Hao Wu","submitted_at":"2013-08-14T15:31:29Z","abstract_excerpt":"We define a homology $\\mathcal{H}_N$ for closed braids by applying Khovanov and Rozansky's matrix factorization construction with potential $ax^{N+1}$. 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