Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo

Flat knot 4.5

Min(phi) over symmetries of the knot is: [-1,-1,1,1,-1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['4.5', '5.97', '6.2072']
Arrow polynomial of the knot is: -4K12 + 2K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['4.5', '4.7', '4.10', '4.11', '6.142', '6.563', '6.606', '6.788', '6.892', '6.944', '6.949', '6.971', '6.1011', '6.1060', '6.1124', '6.1212', '6.1238', '6.1241', '6.1274', '6.1291', '6.1304', '6.1309', '6.1312', '6.1373', '6.1390', '6.1392', '6.1393', '6.1394', '6.1403', '6.1407', '6.1412', '6.1413', '6.1423', '6.1424', '6.1425', '6.1426', '6.1438', '6.1440', '6.1448', '6.1449', '6.1452', '6.1453', '6.1456', '6.1457', '6.1478', '6.1479', '6.1520', '6.1554', '6.1559', '6.1588', '6.1589', '6.1609', '6.1610', '6.1619', '6.1621', '6.1626', '6.1630', '6.1632', '6.1633', '6.1643', '6.1657', '6.1689', '6.1721', '6.1723', '6.1737', '6.1764', '6.1777', '6.1783', '6.1808', '6.1816', '6.1853', '6.1855', '6.1856', '6.1860', '6.1864', '6.1871', '6.1872', '6.1875', '6.1882', '6.1891', '6.1894', '6.1895', '6.1896', '6.1897', '6.1898', '6.1900', '6.1902', '6.1903', '6.1938', '6.1940', '6.1942', '6.1946', '6.1947', '6.1952', '6.1956', '6.1957', '6.1959', '6.1965', '6.1968', '6.1969', '6.1970', '6.1972', '6.1973', '6.1974', '6.2000', '6.2006', '6.2012', '6.2032', '6.2033', '6.2035', '6.2036', '6.2037', '6.2038', '6.2040', '6.2041', '6.2042', '6.2044', '6.2045', '6.2047', '6.2048', '6.2049', '6.2052', '6.2053', '6.2054', '6.2055', '6.2058', '6.2060', '6.2061', '6.2062', '6.2067', '6.2069', '6.2072', '6.2073', '6.2076', '6.2077', '6.2080']
Outer characteristic polynomial of the knot is: t^5+10t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['4.5', '5.97', '6.831', '6.2072', '7.44282']
2-strand cable arrow polynomial of the knot is: -256K14 - 448K1*2K2*2 + 480K1*2K2 - 16K12 + 256K1K2K3 - 176K24 + 96K2*2K4 - 16K32 - 4K4**2 + 82
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['4.5']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk4.2', 'vk4.5', 'vk4.7', 'vk4.56', 'vk4.76', 'vk6.7', 'vk6.10', 'vk6.92', 'vk6.95', 'vk6.108', 'vk6.111', 'vk6.140', 'vk6.143', 'vk6.188', 'vk6.191', 'vk6.204', 'vk6.207', 'vk6.236', 'vk6.239', 'vk6.527', 'vk6.532', 'vk6.535', 'vk6.550', 'vk6.565', 'vk6.646', 'vk6.649', 'vk6.778', 'vk6.814', 'vk6.820', 'vk6.854', 'vk6.906', 'vk6.922', 'vk6.925', 'vk6.930', 'vk6.933', 'vk6.941', 'vk6.951', 'vk6.959', 'vk6.963', 'vk6.971', 'vk6.974', 'vk6.977', 'vk6.1040', 'vk6.1054', 'vk6.1059', 'vk6.1064', 'vk6.1086', 'vk6.1144', 'vk6.1166', 'vk6.1172', 'vk6.1175', 'vk6.1178', 'vk6.1181', 'vk6.1188', 'vk6.1190', 'vk6.1195', 'vk6.1236', 'vk6.1239', 'vk6.1252', 'vk6.1255', 'vk6.1284', 'vk6.1287', 'vk6.1326', 'vk6.1329', 'vk6.1342', 'vk6.1345', 'vk6.1374', 'vk6.1377', 'vk6.1424', 'vk6.1436', 'vk6.1464', 'vk6.1535', 'vk6.1547', 'vk6.1557', 'vk6.1563', 'vk6.1602', 'vk6.1609', 'vk6.1627', 'vk6.1630', 'vk6.1648', 'vk6.1654', 'vk6.1662', 'vk6.1680', 'vk6.1686', 'vk6.1693', 'vk6.1699', 'vk6.1702', 'vk6.1705', 'vk6.1713', 'vk6.1716', 'vk6.1727', 'vk6.1733', 'vk6.1743', 'vk6.1746', 'vk6.1759', 'vk6.1765', 'vk6.1775', 'vk6.1778', 'vk6.1781', 'vk6.1784', 'vk6.1794', 'vk6.1810', 'vk6.1816', 'vk6.1827', 'vk6.1837', 'vk6.1851', 'vk6.1857', 'vk6.1863', 'vk6.1878', 'vk6.1884', 'vk6.1887', 'vk6.1890', 'vk6.1902', 'vk6.1904', 'vk6.1916', 'vk6.2012', 'vk6.2024', 'vk6.2079', 'vk6.2086', 'vk6.2099', 'vk6.2110', 'vk6.2113', 'vk6.2130', 'vk6.2133', 'vk6.2150', 'vk6.2193', 'vk6.2200', 'vk6.2203', 'vk6.2205', 'vk6.2208', 'vk6.2246', 'vk6.2252', 'vk6.2296', 'vk6.2305', 'vk6.2320', 'vk6.2328', 'vk6.2333', 'vk6.2337', 'vk6.2340', 'vk6.2343', 'vk6.2347', 'vk6.2352', 'vk6.2367', 'vk6.2407', 'vk6.2410', 'vk6.2474', 'vk6.2528', 'vk6.2539', 'vk6.2547', 'vk6.2575', 'vk6.2583', 'vk6.2591', 'vk6.2602', 'vk6.2613', 'vk6.2625', 'vk6.2667', 'vk6.2679', 'vk6.2683', 'vk6.2689', 'vk6.2718', 'vk6.2813', 'vk6.2819', 'vk6.2827', 'vk6.2833', 'vk6.2839', 'vk6.2845', 'vk6.2853', 'vk6.2859', 'vk6.2871', 'vk6.2903', 'vk6.2908', 'vk6.2914', 'vk6.2939', 'vk6.2942', 'vk6.3025', 'vk6.3029', 'vk6.3037', 'vk6.3071', 'vk6.3089', 'vk6.3092', 'vk6.3781', 'vk6.3972', 'vk6.10666', 'vk6.10681', 'vk6.10853', 'vk6.10870', 'vk6.12067', 'vk6.13060', 'vk6.14496', 'vk6.14499', 'vk6.14528', 'vk6.14531', 'vk6.14592', 'vk6.14595', 'vk6.14656', 'vk6.14659', 'vk6.14826', 'vk6.14829', 'vk6.14848', 'vk6.14851', 'vk6.15718', 'vk6.15721', 'vk6.15749', 'vk6.15752', 'vk6.15813', 'vk6.15816', 'vk6.15982', 'vk6.15987', 'vk6.16005', 'vk6.16008', 'vk6.21590', 'vk6.21593', 'vk6.21606', 'vk6.21609', 'vk6.21625', 'vk6.21654', 'vk6.21657', 'vk6.21784', 'vk6.21889', 'vk6.22557', 'vk6.22562', 'vk6.24914', 'vk6.25375', 'vk6.25833', 'vk6.25878', 'vk6.25895', 'vk6.25936', 'vk6.25942', 'vk6.25951', 'vk6.25964', 'vk6.25970', 'vk6.26351', 'vk6.26794', 'vk6.27534', 'vk6.27537', 'vk6.27550', 'vk6.27553', 'vk6.27582', 'vk6.27585', 'vk6.27614', 'vk6.27617', 'vk6.27721', 'vk6.27773', 'vk6.27799', 'vk6.27805', 'vk6.27831', 'vk6.27837', 'vk6.27881', 'vk6.27885', 'vk6.27939', 'vk6.27942', 'vk6.27945', 'vk6.28264', 'vk6.28542', 'vk6.28581', 'vk6.28584', 'vk6.29265', 'vk6.29311', 'vk6.29314', 'vk6.29339', 'vk6.29345', 'vk6.29389', 'vk6.29394', 'vk6.29423', 'vk6.29687', 'vk6.29811', 'vk6.29823', 'vk6.30129', 'vk6.30141', 'vk6.30353', 'vk6.30364', 'vk6.30480', 'vk6.30493', 'vk6.30509', 'vk6.30520', 'vk6.30537', 'vk6.30549', 'vk6.30781', 'vk6.30797', 'vk6.30809', 'vk6.31142', 'vk6.31148', 'vk6.31262', 'vk6.31268', 'vk6.31388', 'vk6.31400', 'vk6.31444', 'vk6.31463', 'vk6.31475', 'vk6.31625', 'vk6.31637', 'vk6.31789', 'vk6.31800', 'vk6.31982', 'vk6.31993', 'vk6.32284', 'vk6.32296', 'vk6.32396', 'vk6.32408', 'vk6.32562', 'vk6.32574', 'vk6.32618', 'vk6.32630', 'vk6.32682', 'vk6.32694', 'vk6.32706', 'vk6.32718', 'vk6.32723', 'vk6.32733', 'vk6.32739', 'vk6.32833', 'vk6.32839', 'vk6.32947', 'vk6.32957', 'vk6.32963', 'vk6.32974', 'vk6.33340', 'vk6.33352', 'vk6.33362', 'vk6.33365', 'vk6.33478', 'vk6.33518', 'vk6.33519', 'vk6.33522', 'vk6.33527', 'vk6.33554', 'vk6.33557', 'vk6.33586', 'vk6.33589', 'vk6.34175', 'vk6.34178', 'vk6.34201', 'vk6.34483', 'vk6.36581', 'vk6.38011', 'vk6.38997', 'vk6.39003', 'vk6.39161', 'vk6.39167', 'vk6.39225', 'vk6.39231', 'vk6.39257', 'vk6.39263', 'vk6.39320', 'vk6.39326', 'vk6.39358', 'vk6.39560', 'vk6.39566', 'vk6.39717', 'vk6.40213', 'vk6.40224', 'vk6.41163', 'vk6.41166', 'vk6.41195', 'vk6.41198', 'vk6.41245', 'vk6.41250', 'vk6.41295', 'vk6.41298', 'vk6.41439', 'vk6.41442', 'vk6.41499', 'vk6.41504', 'vk6.41529', 'vk6.41532', 'vk6.41535', 'vk6.41788', 'vk6.41798', 'vk6.41963', 'vk6.41969', 'vk6.42228', 'vk6.42234', 'vk6.43692', 'vk6.45039', 'vk6.45088', 'vk6.45699', 'vk6.45702', 'vk6.45730', 'vk6.45733', 'vk6.45763', 'vk6.45766', 'vk6.46173', 'vk6.46181', 'vk6.46714', 'vk6.46722', 'vk6.46814', 'vk6.46865', 'vk6.46877', 'vk6.46885', 'vk6.46896', 'vk6.46908', 'vk6.46948', 'vk6.46954', 'vk6.46960', 'vk6.51746', 'vk6.51758', 'vk6.51925', 'vk6.51941', 'vk6.51953', 'vk6.52158', 'vk6.52164', 'vk6.52262', 'vk6.52268', 'vk6.52362', 'vk6.52381', 'vk6.52393', 'vk6.52507', 'vk6.52519', 'vk6.52620', 'vk6.52631', 'vk6.52965', 'vk6.52977', 'vk6.53081', 'vk6.53093', 'vk6.53184', 'vk6.53196', 'vk6.53244', 'vk6.53256', 'vk6.53288', 'vk6.53303', 'vk6.53309', 'vk6.53417', 'vk6.53423', 'vk6.53498', 'vk6.53508', 'vk6.53727', 'vk6.53730', 'vk6.53763', 'vk6.53765', 'vk6.54336', 'vk6.54338', 'vk6.54362', 'vk6.54365', 'vk6.54540', 'vk6.54542', 'vk6.57144', 'vk6.57546', 'vk6.57552', 'vk6.58268', 'vk6.58271', 'vk6.58300', 'vk6.58303', 'vk6.58380', 'vk6.58383', 'vk6.58523', 'vk6.58526', 'vk6.58993', 'vk6.58998', 'vk6.61674', 'vk6.61677', 'vk6.61705', 'vk6.61708', 'vk6.61760', 'vk6.61763', 'vk6.61890', 'vk6.61946', 'vk6.61970', 'vk6.61976', 'vk6.61999', 'vk6.66902', 'vk6.67608', 'vk6.67621', 'vk6.67624', 'vk6.69383', 'vk6.69398', 'vk6.69401', 'vk6.82889', 'vk6.82892', 'vk6.83017', 'vk6.83021', 'vk6.83407', 'vk6.83412', 'vk6.83485', 'vk6.83489', 'vk6.83951', 'vk6.83953', 'vk6.83994', 'vk6.83998', 'vk6.84006', 'vk6.85999', 'vk6.86298', 'vk6.86304', 'vk6.86330', 'vk6.86336', 'vk6.87903', 'vk6.88187', 'vk6.88520', 'vk6.88532', 'vk6.88560', 'vk6.88758', 'vk6.88841', 'vk6.88852', 'vk6.89063']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is a.
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

Min(phi) over symmetries of the knot is: [-1,-1,1,1,-1,1,1,1,1,-1]

Flat knots (up to 7 crossings) with same phi are :['4.5', '5.97', '6.2072']

Arrow polynomial of the knot is: -4*K1**2 + 2*K2 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['4.5', '4.7', '4.10', '4.11', '6.142', '6.563', '6.606', '6.788', '6.892', '6.944', '6.949', '6.971', '6.1011', '6.1060', '6.1124', '6.1212', '6.1238', '6.1241', '6.1274', '6.1291', '6.1304', '6.1309', '6.1312', '6.1373', '6.1390', '6.1392', '6.1393', '6.1394', '6.1403', '6.1407', '6.1412', '6.1413', '6.1423', '6.1424', '6.1425', '6.1426', '6.1438', '6.1440', '6.1448', '6.1449', '6.1452', '6.1453', '6.1456', '6.1457', '6.1478', '6.1479', '6.1520', '6.1554', '6.1559', '6.1588', '6.1589', '6.1609', '6.1610', '6.1619', '6.1621', '6.1626', '6.1630', '6.1632', '6.1633', '6.1643', '6.1657', '6.1689', '6.1721', '6.1723', '6.1737', '6.1764', '6.1777', '6.1783', '6.1808', '6.1816', '6.1853', '6.1855', '6.1856', '6.1860', '6.1864', '6.1871', '6.1872', '6.1875', '6.1882', '6.1891', '6.1894', '6.1895', '6.1896', '6.1897', '6.1898', '6.1900', '6.1902', '6.1903', '6.1938', '6.1940', '6.1942', '6.1946', '6.1947', '6.1952', '6.1956', '6.1957', '6.1959', '6.1965', '6.1968', '6.1969', '6.1970', '6.1972', '6.1973', '6.1974', '6.2000', '6.2006', '6.2012', '6.2032', '6.2033', '6.2035', '6.2036', '6.2037', '6.2038', '6.2040', '6.2041', '6.2042', '6.2044', '6.2045', '6.2047', '6.2048', '6.2049', '6.2052', '6.2053', '6.2054', '6.2055', '6.2058', '6.2060', '6.2061', '6.2062', '6.2067', '6.2069', '6.2072', '6.2073', '6.2076', '6.2077', '6.2080']

Outer characteristic polynomial of the knot is: t^5+10t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['4.5', '5.97', '6.831', '6.2072', '7.44282']

2-strand cable arrow polynomial of the knot is: -256*K1**4 - 448*K1**2*K2**2 + 480*K1**2*K2 - 16*K1**2 + 256*K1*K2*K3 - 176*K2**4 + 96*K2**2*K4 - 16*K3**2 - 4*K4**2 + 82

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['4.5']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk4.2', 'vk4.5', 'vk4.7', 'vk4.56', 'vk4.76', 'vk6.7', 'vk6.10', 'vk6.92', 'vk6.95', 'vk6.108', 'vk6.111', 'vk6.140', 'vk6.143', 'vk6.188', 'vk6.191', 'vk6.204', 'vk6.207', 'vk6.236', 'vk6.239', 'vk6.527', 'vk6.532', 'vk6.535', 'vk6.550', 'vk6.565', 'vk6.646', 'vk6.649', 'vk6.778', 'vk6.814', 'vk6.820', 'vk6.854', 'vk6.906', 'vk6.922', 'vk6.925', 'vk6.930', 'vk6.933', 'vk6.941', 'vk6.951', 'vk6.959', 'vk6.963', 'vk6.971', 'vk6.974', 'vk6.977', 'vk6.1040', 'vk6.1054', 'vk6.1059', 'vk6.1064', 'vk6.1086', 'vk6.1144', 'vk6.1166', 'vk6.1172', 'vk6.1175', 'vk6.1178', 'vk6.1181', 'vk6.1188', 'vk6.1190', 'vk6.1195', 'vk6.1236', 'vk6.1239', 'vk6.1252', 'vk6.1255', 'vk6.1284', 'vk6.1287', 'vk6.1326', 'vk6.1329', 'vk6.1342', 'vk6.1345', 'vk6.1374', 'vk6.1377', 'vk6.1424', 'vk6.1436', 'vk6.1464', 'vk6.1535', 'vk6.1547', 'vk6.1557', 'vk6.1563', 'vk6.1602', 'vk6.1609', 'vk6.1627', 'vk6.1630', 'vk6.1648', 'vk6.1654', 'vk6.1662', 'vk6.1680', 'vk6.1686', 'vk6.1693', 'vk6.1699', 'vk6.1702', 'vk6.1705', 'vk6.1713', 'vk6.1716', 'vk6.1727', 'vk6.1733', 'vk6.1743', 'vk6.1746', 'vk6.1759', 'vk6.1765', 'vk6.1775', 'vk6.1778', 'vk6.1781', 'vk6.1784', 'vk6.1794', 'vk6.1810', 'vk6.1816', 'vk6.1827', 'vk6.1837', 'vk6.1851', 'vk6.1857', 'vk6.1863', 'vk6.1878', 'vk6.1884', 'vk6.1887', 'vk6.1890', 'vk6.1902', 'vk6.1904', 'vk6.1916', 'vk6.2012', 'vk6.2024', 'vk6.2079', 'vk6.2086', 'vk6.2099', 'vk6.2110', 'vk6.2113', 'vk6.2130', 'vk6.2133', 'vk6.2150', 'vk6.2193', 'vk6.2200', 'vk6.2203', 'vk6.2205', 'vk6.2208', 'vk6.2246', 'vk6.2252', 'vk6.2296', 'vk6.2305', 'vk6.2320', 'vk6.2328', 'vk6.2333', 'vk6.2337', 'vk6.2340', 'vk6.2343', 'vk6.2347', 'vk6.2352', 'vk6.2367', 'vk6.2407', 'vk6.2410', 'vk6.2474', 'vk6.2528', 'vk6.2539', 'vk6.2547', 'vk6.2575', 'vk6.2583', 'vk6.2591', 'vk6.2602', 'vk6.2613', 'vk6.2625', 'vk6.2667', 'vk6.2679', 'vk6.2683', 'vk6.2689', 'vk6.2718', 'vk6.2813', 'vk6.2819', 'vk6.2827', 'vk6.2833', 'vk6.2839', 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The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is a.

The fillings (up to the first 10) associated to the algebraic genus:

Or click here to check the fillings

invariant	value
Gauss code	O1O2O3O4U3U4U1U2
R3 orbit	{'O1O2O3O4U3U4U1U2'}
R3 orbit length	1
Gauss code of -K	Same
Gauss code of K*	Same
Gauss code of -K*	Same
Diagrammatic symmetry type	a
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 1 -1 1],[ 1 0 1 -1 1],[-1 -1 0 -1 1],[ 1 1 1 0 1],[-1 -1 -1 -1 0]]
Primitive based matrix	[[ 0 1 1 -1 -1],[-1 0 1 -1 -1],[-1 -1 0 -1 -1],[ 1 1 1 0 1],[ 1 1 1 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,1,1,-1,1,1,1,1,-1]
Phi over symmetry	[-1,-1,1,1,-1,1,1,1,1,-1]
Phi of -K	[-1,-1,1,1,-1,1,1,1,1,-1]
Phi of K*	[-1,-1,1,1,-1,1,1,1,1,-1]
Phi of -K*	[-1,-1,1,1,-1,1,1,1,1,-1]
Symmetry type of based matrix	a
u-polynomial	0
Normalized Jones-Krushkal polynomial	4z+9
Enhanced Jones-Krushkal polynomial	4w^2z+9w
Inner characteristic polynomial	t^4+6t^2+1
Outer characteristic polynomial	t^5+10t^3+5t
Flat arrow polynomial	-4K12 + 2K2 + 3
2-strand cable arrow polynomial	-256K14 - 448K1*2K2*2 + 480K1*2K2 - 16K12 + 256K1K2K3 - 176K24 + 96K2*2K4 - 16K32 - 4K4**2 + 82
Genus of based matrix	0
Fillings of based matrix	[[{3, 4}, {1, 2}]]
If K is slice	True

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