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A simple drawing is c-monotone if there is a point\u00a0<jats:italic>O<\/jats:italic> such that each ray emanating from\u00a0<jats:italic>O<\/jats:italic> crosses each edge of the drawing at most once. We introduce a special kind of c-monotone drawings that we call generalized twisted drawings. A c-monotone drawing is generalized twisted if there is a ray emanating from\u00a0<jats:italic>O<\/jats:italic> that crosses all the edges of the drawing. Via this class of drawings, we show that every simple drawing of the complete graph with\u00a0<jats:italic>n<\/jats:italic> vertices contains\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Omega (n^{\\frac{1}{2}})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03a9<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mfrac>\n                        <mml:mn>1<\/mml:mn>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mfrac>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> pairwise disjoint edges and a plane cycle (and hence path) of length\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Omega (\\frac{\\log n }{\\log \\log n})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03a9<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mfrac>\n                      <mml:mrow>\n                        <mml:mo>log<\/mml:mo>\n                        <mml:mi>n<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mo>log<\/mml:mo>\n                        <mml:mo>log<\/mml:mo>\n                        <mml:mi>n<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:mfrac>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Both results improve over best previously published lower bounds. On the way we show several structural results and properties of generalized twisted and c-monotone drawings, some of which we believe to be of independent interest. For example, we show that a drawing <jats:italic>D<\/jats:italic> is c-monotone if there exists a point <jats:italic>O<\/jats:italic> such that no edge of <jats:italic>D<\/jats:italic> is crossed more than once by any ray that emanates from <jats:italic>O<\/jats:italic> and passes through a vertex of\u00a0<jats:italic>D<\/jats:italic>.<\/jats:p>","DOI":"10.1007\/s00454-023-00610-0","type":"journal-article","created":{"date-parts":[[2024,1,3]],"date-time":"2024-01-03T21:13:46Z","timestamp":1704316426000},"page":"40-66","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs"],"prefix":"10.1007","volume":"71","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2364-0583","authenticated-orcid":false,"given":"Oswin","family":"Aichholzer","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6519-1472","authenticated-orcid":false,"given":"Alfredo","family":"Garc\u00eda","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9543-7170","authenticated-orcid":false,"given":"Javier","family":"Tejel","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7166-4467","authenticated-orcid":false,"given":"Birgit","family":"Vogtenhuber","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8553-6661","authenticated-orcid":false,"given":"Alexandra","family":"Weinberger","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,1,3]]},"reference":[{"issue":"2","key":"610_CR1","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1007\/s00373-019-02076-5","volume":"36","author":"B \u00c1brego","year":"2020","unstructured":"\u00c1brego, B., Fern\u00e1ndez-Merchant, S., Sparks, A.: The bipartite-cylindrical crossing number of the complete bipartite graph. 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