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

Flat knot 4.10

Min(phi) over symmetries of the knot is: [-1,-1,1,1,0,0,1,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['4.10', '5.106', '6.2037']
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+6t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['4.10', '5.106', '6.2037']
2-strand cable arrow polynomial of the knot is: -128K14 - 192K1*2K2*2 + 288K1*2K2 - 80K12 + 128K1K2K3 - 48K24 + 32K2*2K4 - 64K22 - 16K3*2 - 4K4**2 + 82
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['4.10']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk4.1', 'vk4.4', 'vk4.8', 'vk4.55', 'vk4.77', 'vk6.3', 'vk6.6', 'vk6.80', 'vk6.83', 'vk6.96', 'vk6.99', 'vk6.128', 'vk6.131', 'vk6.176', 'vk6.179', 'vk6.192', 'vk6.195', 'vk6.224', 'vk6.227', 'vk6.524', 'vk6.528', 'vk6.531', 'vk6.543', 'vk6.558', 'vk6.642', 'vk6.645', 'vk6.785', 'vk6.807', 'vk6.813', 'vk6.858', 'vk6.913', 'vk6.918', 'vk6.921', 'vk6.926', 'vk6.929', 'vk6.934', 'vk6.942', 'vk6.952', 'vk6.956', 'vk6.964', 'vk6.967', 'vk6.970', 'vk6.1031', 'vk6.1048', 'vk6.1053', 'vk6.1063', 'vk6.1077', 'vk6.1139', 'vk6.1165', 'vk6.1168', 'vk6.1171', 'vk6.1177', 'vk6.1182', 'vk6.1186', 'vk6.1191', 'vk6.1232', 'vk6.1235', 'vk6.1248', 'vk6.1251', 'vk6.1280', 'vk6.1283', 'vk6.1322', 'vk6.1325', 'vk6.1338', 'vk6.1341', 'vk6.1370', 'vk6.1373', 'vk6.1423', 'vk6.1435', 'vk6.1473', 'vk6.1534', 'vk6.1546', 'vk6.1550', 'vk6.1556', 'vk6.1595', 'vk6.1617', 'vk6.1619', 'vk6.1625', 'vk6.1639', 'vk6.1645', 'vk6.1655', 'vk6.1679', 'vk6.1684', 'vk6.1685', 'vk6.1690', 'vk6.1692', 'vk6.1695', 'vk6.1698', 'vk6.1704', 'vk6.1709', 'vk6.1712', 'vk6.1720', 'vk6.1726', 'vk6.1739', 'vk6.1742', 'vk6.1752', 'vk6.1758', 'vk6.1768', 'vk6.1771', 'vk6.1774', 'vk6.1783', 'vk6.1793', 'vk6.1803', 'vk6.1809', 'vk6.1826', 'vk6.1836', 'vk6.1850', 'vk6.1856', 'vk6.1862', 'vk6.1877', 'vk6.1880', 'vk6.1883', 'vk6.1889', 'vk6.1893', 'vk6.1907', 'vk6.1919', 'vk6.2015', 'vk6.2027', 'vk6.2076', 'vk6.2097', 'vk6.2101', 'vk6.2106', 'vk6.2109', 'vk6.2126', 'vk6.2129', 'vk6.2149', 'vk6.2194', 'vk6.2196', 'vk6.2199', 'vk6.2206', 'vk6.2211', 'vk6.2247', 'vk6.2253', 'vk6.2298', 'vk6.2307', 'vk6.2327', 'vk6.2329', 'vk6.2334', 'vk6.2339', 'vk6.2345', 'vk6.2348', 'vk6.2370', 'vk6.2411', 'vk6.2414', 'vk6.2463', 'vk6.2524', 'vk6.2526', 'vk6.2538', 'vk6.2546', 'vk6.2574', 'vk6.2582', 'vk6.2586', 'vk6.2611', 'vk6.2616', 'vk6.2628', 'vk6.2670', 'vk6.2682', 'vk6.2692', 'vk6.2698', 'vk6.2710', 'vk6.2814', 'vk6.2820', 'vk6.2835', 'vk6.2838', 'vk6.2846', 'vk6.2852', 'vk6.2854', 'vk6.2860', 'vk6.2872', 'vk6.2905', 'vk6.2911', 'vk6.2916', 'vk6.2951', 'vk6.2954', 'vk6.3021', 'vk6.3036', 'vk6.3046', 'vk6.3078', 'vk6.3093', 'vk6.3096', 'vk6.3780', 'vk6.3973', 'vk6.10665', 'vk6.10682', 'vk6.10854', 'vk6.10869', 'vk6.12068', 'vk6.13059', 'vk6.14484', 'vk6.14487', 'vk6.14516', 'vk6.14519', 'vk6.14580', 'vk6.14583', 'vk6.14644', 'vk6.14647', 'vk6.14825', 'vk6.14830', 'vk6.14836', 'vk6.14839', 'vk6.15231', 'vk6.15706', 'vk6.15709', 'vk6.15737', 'vk6.15740', 'vk6.15801', 'vk6.15804', 'vk6.15983', 'vk6.15986', 'vk6.15993', 'vk6.15996', 'vk6.21586', 'vk6.21589', 'vk6.21602', 'vk6.21605', 'vk6.21622', 'vk6.21650', 'vk6.21653', 'vk6.21777', 'vk6.21882', 'vk6.22558', 'vk6.22561', 'vk6.24919', 'vk6.25382', 'vk6.25880', 'vk6.25893', 'vk6.25934', 'vk6.25944', 'vk6.25949', 'vk6.25955', 'vk6.25961', 'vk6.26356', 'vk6.26801', 'vk6.27530', 'vk6.27533', 'vk6.27546', 'vk6.27549', 'vk6.27578', 'vk6.27581', 'vk6.27610', 'vk6.27613', 'vk6.27710', 'vk6.27764', 'vk6.27790', 'vk6.27796', 'vk6.27822', 'vk6.27828', 'vk6.27874', 'vk6.27878', 'vk6.27932', 'vk6.27935', 'vk6.27938', 'vk6.28253', 'vk6.28535', 'vk6.28580', 'vk6.28585', 'vk6.29256', 'vk6.29309', 'vk6.29332', 'vk6.29338', 'vk6.29383', 'vk6.29388', 'vk6.29422', 'vk6.29678', 'vk6.29810', 'vk6.29822', 'vk6.30128', 'vk6.30140', 'vk6.30352', 'vk6.30365', 'vk6.30481', 'vk6.30492', 'vk6.30507', 'vk6.30518', 'vk6.30536', 'vk6.30548', 'vk6.30779', 'vk6.30789', 'vk6.30796', 'vk6.30808', 'vk6.31141', 'vk6.31147', 'vk6.31261', 'vk6.31267', 'vk6.31385', 'vk6.31397', 'vk6.31441', 'vk6.31452', 'vk6.31462', 'vk6.31474', 'vk6.31624', 'vk6.31636', 'vk6.31792', 'vk6.31984', 'vk6.31996', 'vk6.32285', 'vk6.32297', 'vk6.32397', 'vk6.32409', 'vk6.32563', 'vk6.32575', 'vk6.32619', 'vk6.32631', 'vk6.32683', 'vk6.32695', 'vk6.32707', 'vk6.32719', 'vk6.32722', 'vk6.32734', 'vk6.32740', 'vk6.32834', 'vk6.32840', 'vk6.32949', 'vk6.32965', 'vk6.32976', 'vk6.33339', 'vk6.33351', 'vk6.33358', 'vk6.33361', 'vk6.33476', 'vk6.33517', 'vk6.33521', 'vk6.33524', 'vk6.33550', 'vk6.33553', 'vk6.33582', 'vk6.33585', 'vk6.33878', 'vk6.34171', 'vk6.34174', 'vk6.34197', 'vk6.34200', 'vk6.34479', 'vk6.34482', 'vk6.36572', 'vk6.38013', 'vk6.38996', 'vk6.39002', 'vk6.39152', 'vk6.39158', 'vk6.39216', 'vk6.39222', 'vk6.39248', 'vk6.39254', 'vk6.39312', 'vk6.39318', 'vk6.39351', 'vk6.39559', 'vk6.39565', 'vk6.39716', 'vk6.40210', 'vk6.40222', 'vk6.41159', 'vk6.41162', 'vk6.41191', 'vk6.41194', 'vk6.41244', 'vk6.41247', 'vk6.41291', 'vk6.41294', 'vk6.41437', 'vk6.41492', 'vk6.41498', 'vk6.41522', 'vk6.41525', 'vk6.41528', 'vk6.41787', 'vk6.41797', 'vk6.41956', 'vk6.41962', 'vk6.42229', 'vk6.42233', 'vk6.42248', 'vk6.43683', 'vk6.45038', 'vk6.45101', 'vk6.45695', 'vk6.45698', 'vk6.45726', 'vk6.45729', 'vk6.45765', 'vk6.45768', 'vk6.46174', 'vk6.46182', 'vk6.46715', 'vk6.46723', 'vk6.46819', 'vk6.46866', 'vk6.46878', 'vk6.46884', 'vk6.46897', 'vk6.46909', 'vk6.46947', 'vk6.46955', 'vk6.46957', 'vk6.46973', 'vk6.51745', 'vk6.51757', 'vk6.51923', 'vk6.51933', 'vk6.51940', 'vk6.51952', 'vk6.52157', 'vk6.52163', 'vk6.52261', 'vk6.52267', 'vk6.52359', 'vk6.52370', 'vk6.52380', 'vk6.52392', 'vk6.52506', 'vk6.52518', 'vk6.52623', 'vk6.52966', 'vk6.52978', 'vk6.53082', 'vk6.53094', 'vk6.53185', 'vk6.53197', 'vk6.53245', 'vk6.53257', 'vk6.53289', 'vk6.53304', 'vk6.53310', 'vk6.53418', 'vk6.53424', 'vk6.53500', 'vk6.53731', 'vk6.53734', 'vk6.53762', 'vk6.54339', 'vk6.54366', 'vk6.54369', 'vk6.54453', 'vk6.54543', 'vk6.57141', 'vk6.57545', 'vk6.57551', 'vk6.58272', 'vk6.58275', 'vk6.58304', 'vk6.58307', 'vk6.58384', 'vk6.58387', 'vk6.58527', 'vk6.58530', 'vk6.58992', 'vk6.58997', 'vk6.61678', 'vk6.61681', 'vk6.61709', 'vk6.61712', 'vk6.61761', 'vk6.61766', 'vk6.61891', 'vk6.61947', 'vk6.61977', 'vk6.61983', 'vk6.62002', 'vk6.62004', 'vk6.66905', 'vk6.66907', 'vk6.67609', 'vk6.67612', 'vk6.67625', 'vk6.67628', 'vk6.69386', 'vk6.69389', 'vk6.69402', 'vk6.69405', 'vk6.72261', 'vk6.82891', 'vk6.83019', 'vk6.83411', 'vk6.83490', 'vk6.83495', 'vk6.83954', 'vk6.83993', 'vk6.83997', 'vk6.84005', 'vk6.84009', 'vk6.86306', 'vk6.86312', 'vk6.86340', 'vk6.86959', 'vk6.87171', 'vk6.87776', 'vk6.87902', 'vk6.87906', 'vk6.88005', 'vk6.88519', 'vk6.88559', 'vk6.88572', 'vk6.88842', 'vk6.89072']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The mirror image K* is
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,-1,1,1,0,0,1,1,0,0]

Flat knots (up to 7 crossings) with same phi are :['4.10', '5.106', '6.2037']

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+6t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['4.10', '5.106', '6.2037']

2-strand cable arrow polynomial of the knot is: -128*K1**4 - 192*K1**2*K2**2 + 288*K1**2*K2 - 80*K1**2 + 128*K1*K2*K3 - 48*K2**4 + 32*K2**2*K4 - 64*K2**2 - 16*K3**2 - 4*K4**2 + 82

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['4.10']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk4.1', 'vk4.4', 'vk4.8', 'vk4.55', 'vk4.77', 'vk6.3', 'vk6.6', 'vk6.80', 'vk6.83', 'vk6.96', 'vk6.99', 'vk6.128', 'vk6.131', 'vk6.176', 'vk6.179', 'vk6.192', 'vk6.195', 'vk6.224', 'vk6.227', 'vk6.524', 'vk6.528', 'vk6.531', 'vk6.543', 'vk6.558', 'vk6.642', 'vk6.645', 'vk6.785', 'vk6.807', 'vk6.813', 'vk6.858', 'vk6.913', 'vk6.918', 'vk6.921', 'vk6.926', 'vk6.929', 'vk6.934', 'vk6.942', 'vk6.952', 'vk6.956', 'vk6.964', 'vk6.967', 'vk6.970', 'vk6.1031', 'vk6.1048', 'vk6.1053', 'vk6.1063', 'vk6.1077', 'vk6.1139', 'vk6.1165', 'vk6.1168', 'vk6.1171', 'vk6.1177', 'vk6.1182', 'vk6.1186', 'vk6.1191', 'vk6.1232', 'vk6.1235', 'vk6.1248', 'vk6.1251', 'vk6.1280', 'vk6.1283', 'vk6.1322', 'vk6.1325', 'vk6.1338', 'vk6.1341', 'vk6.1370', 'vk6.1373', 'vk6.1423', 'vk6.1435', 'vk6.1473', 'vk6.1534', 'vk6.1546', 'vk6.1550', 'vk6.1556', 'vk6.1595', 'vk6.1617', 'vk6.1619', 'vk6.1625', 'vk6.1639', 'vk6.1645', 'vk6.1655', 'vk6.1679', 'vk6.1684', 'vk6.1685', 'vk6.1690', 'vk6.1692', 'vk6.1695', 'vk6.1698', 'vk6.1704', 'vk6.1709', 'vk6.1712', 'vk6.1720', 'vk6.1726', 'vk6.1739', 'vk6.1742', 'vk6.1752', 'vk6.1758', 'vk6.1768', 'vk6.1771', 'vk6.1774', 'vk6.1783', 'vk6.1793', 'vk6.1803', 'vk6.1809', 'vk6.1826', 'vk6.1836', 'vk6.1850', 'vk6.1856', 'vk6.1862', 'vk6.1877', 'vk6.1880', 'vk6.1883', 'vk6.1889', 'vk6.1893', 'vk6.1907', 'vk6.1919', 'vk6.2015', 'vk6.2027', 'vk6.2076', 'vk6.2097', 'vk6.2101', 'vk6.2106', 'vk6.2109', 'vk6.2126', 'vk6.2129', 'vk6.2149', 'vk6.2194', 'vk6.2196', 'vk6.2199', 'vk6.2206', 'vk6.2211', 'vk6.2247', 'vk6.2253', 'vk6.2298', 'vk6.2307', 'vk6.2327', 'vk6.2329', 'vk6.2334', 'vk6.2339', 'vk6.2345', 'vk6.2348', 'vk6.2370', 'vk6.2411', 'vk6.2414', 'vk6.2463', 'vk6.2524', 'vk6.2526', 'vk6.2538', 'vk6.2546', 'vk6.2574', 'vk6.2582', 'vk6.2586', 'vk6.2611', 'vk6.2616', 'vk6.2628', 'vk6.2670', 'vk6.2682', 'vk6.2692', 'vk6.2698', 'vk6.2710', 'vk6.2814', 'vk6.2820', 'vk6.2835', 'vk6.2838', 'vk6.2846', 'vk6.2852', 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The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is -.

The reverse -K is

The 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	O1O2U1U2O3O4U3U4
R3 orbit	{'O1O2U1U2O3O4U3U4'}
R3 orbit length	1
Gauss code of -K	O1O2U3U4O3O4U1U2
Gauss code of K*	O1O2U3U4O3O4U1U2
Gauss code of -K*	Same
Diagrammatic symmetry type	-
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 0 0],[-1 -1 0 0 0],[ 1 0 0 0 1],[-1 0 0 -1 0]]
Primitive based matrix	[[ 0 1 1 -1 -1],[-1 0 0 0 -1],[-1 0 0 -1 0],[ 1 0 1 0 0],[ 1 1 0 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,1,1,0,0,1,1,0,0]
Phi over symmetry	[-1,-1,1,1,0,0,1,1,0,0]
Phi of -K	[-1,-1,1,1,0,1,2,2,1,0]
Phi of K*	[-1,-1,1,1,0,1,2,2,1,0]
Phi of -K*	[-1,-1,1,1,0,0,1,1,0,0]
Symmetry type of based matrix	-
u-polynomial	0
Normalized Jones-Krushkal polynomial	4z+9
Enhanced Jones-Krushkal polynomial	4w^2z+9w
Inner characteristic polynomial	t^4+2t^2+1
Outer characteristic polynomial	t^5+6t^3+5t
Flat arrow polynomial	-4K12 + 2K2 + 3
2-strand cable arrow polynomial	-128K14 - 192K1*2K2*2 + 288K1*2K2 - 80K12 + 128K1K2K3 - 48K24 + 32K2*2K4 - 64K22 - 16K3*2 - 4K4**2 + 82
Genus of based matrix	0
Fillings of based matrix	[[{1, 4}, {2, 3}], [{3, 4}, {1, 2}]]
If K is slice	True

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