{"paper":{"title":"Mobius transformations of polygons and partitions of 3-space","arxiv_id":"math/0602466","primary_cat":"math.GT","submitted_at":"2006-02-21T19:30:26+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.9,"id":"s0","own":false,"role":"background","text":"The image of a polygonal knot K under a spherical inversion of R^3 (union infinity) is a simple closed curve made of arcs of circles, having the same knot type as the mirror image of K.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.2,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.2,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.1,"background":0.9,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":0.87,"id":"s1","own":false,"role":"method","text":"Suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon.","hedge":{"p":0.5,"label":"asserted"},"own_p":0.27,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.87,"result":0.0,"background":0.08,"limitation":0.0,"future_work":0.0,"contribution":0.05,"interpretation":0.0}},{"i":2,"p":0.46,"id":"s2","own":false,"role":"background","text":"This may be a different knot type.","hedge":{"p":1.0,"label":"hedged_possibility"},"own_p":0.19,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.02,"method":0.01,"result":0.41,"background":0.46,"limitation":0.01,"future_work":0.0,"contribution":0.01,"interpretation":0.08}},{"i":3,"p":0.76,"id":"s3","own":true,"role":"result","text":"For example, a certain 7-segment figure-8 knot can be transformed to a figure-8 knot, right and left handed trefoils, or an unknot, by selecting different inverting spheres.","hedge":{"p":0.92,"label":"asserted"},"own_p":0.64,"main_claim_p":0.05,"quantitative":true,"quantitative_p":0.7,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.76,"background":0.2,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.03}},{"i":4,"p":0.54,"id":"s4","own":false,"role":"gap","text":"Which knot types can be obtained from a given original polygon K under this process?","hedge":{"p":0.95,"label":"unclear"},"own_p":0.07,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.54,"method":0.0,"result":0.0,"background":0.45,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":5,"p":0.99,"id":"s5","own":true,"role":"result","text":"We show that for large n, almost all n-segment knot types cannot be reached from one initial n-segment polygon, using a single inversion or even the whole Mobius group.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.72,"quantitative":true,"quantitative_p":0.8,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":6,"p":0.91,"id":"s6","own":true,"role":"result","text":"The number of knot types arising from an n-vertex polygon is bounded by the number of complementary domains of a certain system of n(n-3)/2 round 2-spheres in R^3.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.86,"main_claim_p":0.18,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.91,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.07,"interpretation":0.0}},{"i":7,"p":0.98,"id":"s7","own":true,"role":"result","text":"We show the number of domains is at most polynomial in the number of spheres.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.11,"quantitative":true,"quantitative_p":0.95,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.98,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}},{"i":8,"p":0.85,"id":"s8","own":true,"role":"result","text":"In the analysis, we obtain an exact formula for the number of complementary domains.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.23,"role_probabilities":{"gap":0.0,"method":0.03,"result":0.85,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.12,"interpretation":0.0}},{"i":9,"p":0.65,"id":"s9","own":false,"role":"result","text":"On the other hand, the number of knot types that can be represented by n-segment polygons is exponential in 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