{"paper":{"title":"Catalan States of Lattice Crossing: Application of Plucking Polynomial","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jozef H. Przytycki, Mieczyslaw K. Dabkowski","submitted_at":"2017-11-14T22:06:33Z","abstract_excerpt":"For a Catalan state $C$ of a lattice crossing $L\\left( m,n\\right) $ with no returns on one side, we find its coefficient $C\\left( A\\right) $ in the Relative Kauffman Bracket Skein Module expansion of $L\\left( m,n\\right) $. We show, in particular, that $C\\left( A\\right) $ can be found using the plucking polynomial of a rooted tree with a delay function associated to $C$. Furthermore, for $C$ with returns on one side only, we prove that $C\\left( A\\right) $ is a product of Gaussian polynomials, and its coefficients form a unimodal sequence."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.05328","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}