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Flat knot 5.49

Min(phi) over symmetries of the knot is: [-2,-1,0,1,2,-1,0,2,3,0,1,1,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['5.49', '7.27100']
Arrow polynomial of the knot is: 8*K1**2 + 4*K1*K2 - 2*K1 - 4*K2 - 2*K3 - 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['5.19', '5.49', '5.60', '7.4985', '7.5017', '7.5137', '7.6001', '7.6069', '7.6455', '7.6791', '7.7930', '7.8148', '7.8628', '7.8672', '7.8844', '7.8911', '7.8956', '7.10094', '7.10145', '7.10783', '7.11066', '7.11080', '7.11100', '7.11310', '7.11367', '7.11389', '7.11711', '7.11839', '7.11843', '7.11961', '7.11975', '7.12120', '7.12185', '7.12269', '7.12377', '7.12380', '7.12506', '7.12747', '7.12957', '7.12964', '7.13017', '7.13685', '7.13928', '7.14828', '7.15561', '7.15649', '7.15684', '7.15711', '7.15866', '7.15964', '7.16054', '7.16058', '7.16140', '7.16141', '7.16165', '7.16166', '7.16179', '7.16202', '7.16492', '7.16672', '7.16801', '7.16925', '7.17046', '7.17054', '7.17105', '7.17194', '7.17285', '7.17286', '7.17322', '7.17790', '7.17820', '7.18031', '7.18870', '7.18948', '7.19158', '7.19175', '7.19180', '7.19353', '7.19354', '7.19673', '7.19993', '7.20033', '7.20042', '7.20123', '7.20127', '7.20212', '7.20340', '7.20354', '7.20758', '7.20769', '7.21319', '7.21352', '7.21433', '7.21457', '7.21547', '7.21578', '7.21733', '7.21762', '7.21780', '7.22145', '7.22149', '7.22366', '7.22648', '7.22859', '7.23210', '7.23383', '7.24221', '7.24259', '7.24324', '7.24422', '7.24432', '7.24463', '7.24469', '7.24498', '7.24532', '7.24651', '7.24740', '7.24822', '7.25041', '7.25114', '7.25429', '7.25430', '7.25489', '7.25528', '7.25601', '7.25625', '7.25658', '7.25665', '7.25670', '7.25676', '7.25699', '7.26182', '7.26200', '7.26500', '7.26641', '7.26648', '7.26757', '7.27103', '7.27135', '7.27319', '7.27400', '7.27521', '7.27522', '7.27523', '7.27530', '7.27537', '7.27642', '7.27648', '7.27654', '7.27662', '7.27692', '7.27767', '7.27781', '7.27884', '7.27886', '7.27889', '7.27890', '7.27954', '7.28163', '7.28233', '7.28248', '7.28313', '7.28317', '7.28339', '7.28377', '7.28394', '7.28442', '7.28444', '7.28472', '7.28552', '7.28606', '7.28610', '7.28761', '7.28825', '7.28868', '7.29011', '7.29562', '7.29626', '7.29697', '7.29751', '7.29771', '7.29959', '7.30393', '7.30434', '7.30442', '7.30470', '7.30475', '7.30493', '7.30676', '7.30735', '7.30803', '7.30897', '7.30910', '7.31050', '7.31191', '7.31257', '7.31261', '7.31453', '7.31530', '7.31537', '7.31729', '7.31810', '7.32212', '7.32217', '7.32242', '7.32264', '7.32383', '7.32411', '7.32471', '7.32475', '7.32497', '7.32588', '7.32745', '7.32957', '7.33086', '7.33133', '7.33340', '7.33442', '7.33480', '7.33537', '7.33580', '7.33675', '7.33718', '7.33976', '7.34125', '7.34179', '7.34280', '7.34356', '7.34425', '7.34722', '7.34757', '7.34758', '7.34946', '7.34947', '7.34953', '7.34972', '7.35005', '7.35044', '7.35073', '7.35164', '7.35176', '7.35726', '7.35749', '7.35755', '7.35914', '7.35920', '7.36001', '7.36045', '7.36153', '7.36263', '7.36290', '7.36741', '7.36805', '7.36846', '7.36859', '7.36882', '7.36908', '7.36977', '7.37124', '7.37127', '7.37143', '7.37146', '7.37157', '7.37244', '7.37277', '7.37362', '7.37363', '7.37369', '7.37373', '7.38103', '7.38110', '7.38154', '7.38207', '7.38234', '7.38288', '7.38328', '7.38418', '7.38430', '7.38442', '7.38480', '7.38510', '7.38668', '7.38682', '7.38705', '7.38864', '7.38957', '7.39299', '7.39314', '7.39316', '7.39319', '7.39327', '7.39337', '7.39534', '7.39538', '7.39539', '7.39549', '7.39552', '7.39732', '7.39744', '7.39756', '7.39770', '7.39801', '7.39823', '7.39901', '7.39910', '7.39963', '7.40028', '7.40068', '7.40069', '7.40138', '7.40323', '7.40324', '7.40325', '7.40357', '7.40403', '7.40442', '7.40489', '7.40505', '7.40506', '7.40507', '7.40574', '7.40601', '7.40620', '7.40813', '7.40834', '7.42487', '7.42644', '7.42692', '7.42731', '7.42794', '7.43845', '7.43860', '7.43928', '7.43939', '7.44100', '7.44223']
Outer characteristic polynomial of the knot is: t^6+27t^4+19t^2
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.49', '7.35704', '7.40541']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2 - 816*K1**4 + 64*K1**3*K2*K3 - 656*K1**2*K2**2 + 1432*K1**2*K2 - 208*K1**2*K3**2 - 32*K1**2*K4**2 - 968*K1**2 + 1136*K1*K2*K3 + 264*K1*K3*K4 + 56*K1*K4*K5 - 64*K2**4 - 112*K2**2*K3**2 - 48*K2**2*K4**2 + 208*K2**2*K4 - 980*K2**2 + 104*K2*K3*K5 + 32*K2*K4*K6 - 476*K3**2 - 180*K4**2 - 44*K5**2 - 4*K6**2 + 1034
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.49']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.183', 'vk5.192', 'vk5.232', 'vk5.239', 'vk5.322', 'vk5.331', 'vk5.370', 'vk5.670', 'vk5.674', 'vk5.821', 'vk5.1332', 'vk5.1420', 'vk5.1429', 'vk5.1451', 'vk5.1683', 'vk5.1695']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3O4U2U4O5U3U1U5
R3 orbit {'O1O2O3O4U2U4O5U3U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4U2O5U1U3
Gauss code of K* O1O2O3U2U4U1U5O4O5U3
Gauss code of -K* O1O2O3U1O4O5U4U3U5U2
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 -2 0 1 2],[ 1 0 -2 1 1 2],[ 2 2 0 2 1 1],[ 0 -1 -2 0 0 1],[-1 -1 -1 0 0 0],[-2 -2 -1 -1 0 0]]
Primitive based matrix [[ 0 2 1 0 -1 -2],[-2 0 0 -1 -2 -1],[-1 0 0 0 -1 -1],[ 0 1 0 0 -1 -2],[ 1 2 1 1 0 -2],[ 2 1 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,2,0,1,2,1,0,1,1,1,2,2]
Phi over symmetry [-2,-1,0,1,2,-1,0,2,3,0,1,1,1,1,1]
Phi of -K [-2,-1,0,1,2,-1,0,2,3,0,1,1,1,1,1]
Phi of K* [-2,-1,0,1,2,1,1,1,3,1,1,2,0,0,-1]
Phi of -K* [-2,-1,0,1,2,2,2,1,1,1,1,2,0,1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial -10z-19
Enhanced Jones-Krushkal polynomial -10w^2z-19w
Inner characteristic polynomial t^5+17t^3+5t
Outer characteristic polynomial t^6+27t^4+19t^2
Flat arrow polynomial 8*K1**2 + 4*K1*K2 - 2*K1 - 4*K2 - 2*K3 - 3
2-strand cable arrow polynomial 256*K1**4*K2 - 816*K1**4 + 64*K1**3*K2*K3 - 656*K1**2*K2**2 + 1432*K1**2*K2 - 208*K1**2*K3**2 - 32*K1**2*K4**2 - 968*K1**2 + 1136*K1*K2*K3 + 264*K1*K3*K4 + 56*K1*K4*K5 - 64*K2**4 - 112*K2**2*K3**2 - 48*K2**2*K4**2 + 208*K2**2*K4 - 980*K2**2 + 104*K2*K3*K5 + 32*K2*K4*K6 - 476*K3**2 - 180*K4**2 - 44*K5**2 - 4*K6**2 + 1034
Genus of based matrix 1
Fillings of based matrix [[{1, 5}, {4}, {2, 3}], [{2, 5}, {1, 4}, {3}], [{3, 5}, {1, 4}, {2}], [{4, 5}, {2, 3}, {1}], [{5}, {1, 4}, {2, 3}], [{5}, {4}, {2, 3}, {1}]]
If K is slice False
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