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We provide a complete facet description of this polytope, characterize its edges and show that it is Hirsch. We also show that connected blocks polytopes admit a regular unimodular triangulation by constructing a squarefree Gr\u00f6bner basis. In addition, we prove that the polytope is Gorenstein of index 2 and that its <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$h^*$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>h<\/mml:mi>\n                    <mml:mo>\u2217<\/mml:mo>\n                  <\/mml:msup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-vector is unimodal.<\/jats:p>","DOI":"10.1007\/s00454-024-00675-5","type":"journal-article","created":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T14:03:22Z","timestamp":1720706602000},"page":"946-972","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Connected Blocks Polytope"],"prefix":"10.1007","volume":"73","author":[{"given":"Justus","family":"Bruckamp","sequence":"first","affiliation":[]},{"given":"Markus","family":"Chimani","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1417-2722","authenticated-orcid":false,"given":"Martina","family":"Juhnke","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,7,11]]},"reference":[{"issue":"1","key":"675_CR1","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1016\/0095-8956(86)90063-8","volume":"40","author":"F Barahona","year":"1986","unstructured":"Barahona, F., Gr\u00f6tschel, M.: On the cycle polytope of a binary matroid. 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