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Flat knot 5.17

Min(phi) over symmetries of the knot is: [-3,-1,0,2,2,0,2,2,3,1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['5.17', '7.10066', '7.21953', '7.31851']
Arrow polynomial of the knot is: 4*K1**2 + 2*K1*K2 - K1 - 2*K2 - K3 - 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['5.17', '5.30', '5.43', '5.44', '5.61', '7.1720', '7.2276', '7.2750', '7.3797', '7.4196', '7.4225', '7.4532', '7.4713', '7.6161', '7.7507', '7.7519', '7.7561', '7.8044', '7.8477', '7.8490', '7.8902', '7.9088', '7.9112', '7.9114', '7.9122', '7.9254', '7.9678', '7.9914', '7.9916', '7.9944', '7.9946', '7.10083', '7.10181', '7.10244', '7.10253', '7.10340', '7.10629', '7.10749', '7.10796', '7.10827', '7.10832', '7.10846', '7.10939', '7.10960', '7.11263', '7.11316', '7.11476', '7.11719', '7.11723', '7.11726', '7.11869', '7.12035', '7.12050', '7.12137', '7.12144', '7.12154', '7.12530', '7.12781', '7.12855', '7.12963', '7.12978', '7.13015', '7.13430', '7.13543', '7.13719', '7.13721', '7.13818', '7.13853', '7.13969', '7.14071', '7.14102', '7.14106', '7.14152', '7.14207', '7.14236', '7.14771', '7.15101', '7.15104', '7.15110', '7.15120', '7.15132', '7.15394', '7.15410', '7.15460', '7.15633', '7.15778', '7.15784', '7.15836', '7.15916', '7.15980', '7.15982', '7.16044', '7.16048', '7.16114', '7.16118', '7.16124', '7.16145', '7.16147', '7.16149', '7.16164', '7.16174', '7.16180', '7.16249', '7.16252', '7.16273', '7.16293', '7.16566', '7.16569', '7.16989', '7.16995', '7.17095', '7.17096', '7.17106', '7.17115', '7.17160', '7.17162', '7.17178', '7.17328', '7.17348', '7.17608', '7.17625', '7.17698', '7.17745', '7.18123', '7.18125', '7.18127', '7.18175', '7.18177', '7.18325', '7.18367', '7.18371', '7.18373', '7.18374', '7.18393', '7.18395', '7.18399', '7.18483', '7.18488', '7.18495', '7.18497', '7.18514', '7.18574', '7.18579', '7.18587', '7.18588', '7.18669', '7.18696', '7.18814', '7.18908', '7.18914', '7.18915', '7.18916', '7.18925', '7.18927', '7.18934', '7.18940', '7.19021', '7.19026', '7.19033', '7.19034', '7.19047', '7.19054', '7.19111', '7.19123', '7.19124', '7.19346', '7.19364', '7.19398', '7.19414', '7.19416', '7.19439', '7.19443', '7.19457', '7.19483', '7.19485', '7.19567', '7.19570', '7.19616', '7.19638', '7.19677', '7.19843', '7.19847', '7.19867', '7.19912', '7.19951', '7.20094', '7.20148', '7.20221', '7.20223', '7.20225', '7.20226', '7.20428', '7.20508', '7.20509', '7.20514', '7.20517', '7.20521', '7.20524', '7.20911', '7.20937', '7.20943', '7.20950', '7.20951', '7.20999', '7.21049', '7.21050', '7.21051', '7.21054', '7.21056', '7.21059', '7.21063', '7.21088', '7.21309', '7.21311', '7.21313', '7.21317', '7.21366', '7.21372', '7.21376', '7.21383', '7.21390', '7.21398', '7.21437', '7.21449', '7.21463', '7.21740', '7.21763', '7.21801', '7.21807', '7.21827', '7.21831', '7.21834', '7.21853', '7.21923', '7.21930', '7.21934', '7.22004', '7.22028', '7.22045', '7.22183', '7.22192', '7.22276', '7.22282', '7.22493', '7.22572', '7.22622', '7.22688', '7.22772', '7.22781', '7.22951', '7.22952', '7.22967', '7.23021', '7.23307', '7.23309', '7.23350', '7.23493', '7.23506', '7.23635', '7.23636', '7.23728', '7.23802', '7.23920', '7.23969', '7.23995', '7.24032', '7.24095', '7.24148', '7.24150', '7.24165', '7.24173', '7.24338', '7.24362', '7.24364', '7.24505', '7.24558', '7.24662', '7.24852', '7.24891', '7.24896', '7.24981', '7.24985', '7.25165', '7.25167', '7.25171', '7.25173', '7.25209', '7.25296', '7.25337', '7.25352', '7.25517', '7.25518', '7.25527', '7.25539', '7.25540', '7.26104', '7.26112', '7.26213', '7.26300', '7.26504', '7.26567', '7.26809', '7.26826', '7.28032', '7.28087', '7.28653', '7.28749', '7.29006', '7.29091', '7.29102', '7.29180', '7.29286', '7.29443', '7.29451', '7.29848', '7.30042', '7.30362', '7.31033', '7.31058', '7.31219', '7.31220', '7.31450', '7.31451', '7.31489', '7.31495', '7.31541', '7.31556', '7.31618', '7.31619', '7.31632', '7.31645', '7.31663', '7.31679', '7.31680', '7.31682', '7.31684', '7.31718', '7.31720', '7.31721', '7.31730', '7.31773', '7.31824', '7.31834', '7.31835', '7.31842', '7.31885', '7.32101', '7.32103', '7.32127', '7.32141', '7.32157', '7.32163', '7.32871', '7.32987', '7.33097', '7.33105', '7.33108', '7.33132', '7.33169', '7.33243', '7.33276', '7.33312', '7.33366', '7.33368', '7.33452', '7.33458', '7.33472', '7.33474', '7.33475', '7.33477', '7.33486', '7.33507', '7.33509', '7.33768', '7.33987', '7.34003', '7.34010', '7.34453']
Outer characteristic polynomial of the knot is: t^6+47t^4+8t^2
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.17', '7.10066', '7.31851']
2-strand cable arrow polynomial of the knot is: -48*K1**2*K2**2 + 176*K1**2*K2 - 32*K1**2*K3**2 - 276*K1**2 + 264*K1*K2*K3 + 48*K1*K3*K4 + 8*K1*K4*K5 - 16*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 32*K2**2*K4 - 198*K2**2 + 24*K2*K3*K5 + 8*K2*K4*K6 - 124*K3**2 - 32*K4**2 - 8*K5**2 - 2*K6**2 + 214
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.17']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.164', 'vk5.199', 'vk5.292', 'vk5.351', 'vk5.658', 'vk5.805', 'vk5.1181', 'vk5.1324', 'vk5.1377', 'vk5.1394', 'vk5.1409', 'vk5.1436', 'vk5.1667', 'vk5.1787', 'vk5.1885', 'vk5.1945', 'vk5.1975', 'vk5.1996', 'vk5.2095', 'vk5.2101', 'vk5.2175', 'vk5.2309', 'vk5.2323', 'vk5.2405']
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 O1O2O3O4O5U2U4U5U1U3
R3 orbit {'O1O2O3O4U1U3O5U2U4U5', 'O1O2O3O4U1O5U4U2U5U3', 'O1O2O3O4O5U2U4U5U1U3'}
R3 orbit length 3
Gauss code of -K O1O2O3O4O5U3U5U1U2U4
Gauss code of K* O1O2O3O4O5U4U1U5U2U3
Gauss code of -K* O1O2O3O4O5U3U4U1U5U2
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 -3 2 0 2],[ 1 0 -2 2 0 2],[ 3 2 0 3 1 2],[-2 -2 -3 0 -1 1],[ 0 0 -1 1 0 1],[-2 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 2 2 0 -1 -3],[-2 0 1 -1 -2 -3],[-2 -1 0 -1 -2 -2],[ 0 1 1 0 0 -1],[ 1 2 2 0 0 -2],[ 3 3 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,1,3,-1,1,2,3,1,2,2,0,1,2]
Phi over symmetry [-3,-1,0,2,2,0,2,2,3,1,1,1,1,1,-1]
Phi of -K [-3,-1,0,2,2,0,2,2,3,1,1,1,1,1,-1]
Phi of K* [-2,-2,0,1,3,-1,1,1,3,1,1,2,1,2,0]
Phi of -K* [-3,-1,0,2,2,2,1,2,3,0,2,2,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial -5z-9
Enhanced Jones-Krushkal polynomial -5w^2z-9w
Inner characteristic polynomial t^5+29t^3
Outer characteristic polynomial t^6+47t^4+8t^2
Flat arrow polynomial 4*K1**2 + 2*K1*K2 - K1 - 2*K2 - K3 - 1
2-strand cable arrow polynomial -48*K1**2*K2**2 + 176*K1**2*K2 - 32*K1**2*K3**2 - 276*K1**2 + 264*K1*K2*K3 + 48*K1*K3*K4 + 8*K1*K4*K5 - 16*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 32*K2**2*K4 - 198*K2**2 + 24*K2*K3*K5 + 8*K2*K4*K6 - 124*K3**2 - 32*K4**2 - 8*K5**2 - 2*K6**2 + 214
Genus of based matrix 1
Fillings of based matrix [[{2, 5}, {3, 4}, {1}], [{4, 5}, {3}, {1, 2}], [{5}, {1, 4}, {2, 3}], [{5}, {2, 4}, {1, 3}]]
If K is slice False
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