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

Flat knot 4.6

Min(phi) over symmetries of the knot is: [-2,0,1,1,1,1,2,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['4.6', '6.1770', '6.1951', '7.43053', '7.44462']
Arrow polynomial of the knot is: -2K1*2 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['4.6', '4.8', '6.780', '6.804', '6.914', '6.931', '6.946', '6.960', '6.1002', '6.1016', '6.1019', '6.1051', '6.1058', '6.1078', '6.1102', '6.1115', '6.1217', '6.1294', '6.1306', '6.1317', '6.1321', '6.1324', '6.1336', '6.1377', '6.1416', '6.1420', '6.1427', '6.1429', '6.1434', '6.1436', '6.1437', '6.1439', '6.1441', '6.1444', '6.1450', '6.1451', '6.1458', '6.1459', '6.1477', '6.1482', '6.1490', '6.1503', '6.1504', '6.1511', '6.1521', '6.1547', '6.1560', '6.1561', '6.1562', '6.1597', '6.1598', '6.1600', '6.1601', '6.1608', '6.1620', '6.1622', '6.1624', '6.1634', '6.1635', '6.1637', '6.1638', '6.1713', '6.1725', '6.1758', '6.1846', '6.1933', '6.1944', '6.1949', '6.1950', '6.1951']
Outer characteristic polynomial of the knot is: t^5+15t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['4.6', '7.43053', '7.44462']
2-strand cable arrow polynomial of the knot is: -32K14 - 48K1*2K2*2 + 128K1*2K2 - 84K12 + 48K1K2K3 - 8K24 + 8K2*2K4 - 56K22 - 12K3*2 - 2K4**2 + 56
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['4.6']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk4.10', 'vk4.15', 'vk4.20', 'vk4.23', 'vk4.32', 'vk4.35', 'vk4.38', 'vk4.50', 'vk4.63', 'vk4.67', 'vk6.3527', 'vk6.3530', 'vk6.3551', 'vk6.3554', 'vk6.3675', 'vk6.3678', 'vk6.3699', 'vk6.3702', 'vk6.3707', 'vk6.3710', 'vk6.3731', 'vk6.3734', 'vk6.3771', 'vk6.3774', 'vk6.3795', 'vk6.3798', 'vk6.3867', 'vk6.3870', 'vk6.3891', 'vk6.3894', 'vk6.3899', 'vk6.3902', 'vk6.3923', 'vk6.3926', 'vk6.3963', 'vk6.3966', 'vk6.3987', 'vk6.3990', 'vk6.4898', 'vk6.4910', 'vk6.4915', 'vk6.4918', 'vk6.4939', 'vk6.4942', 'vk6.4963', 'vk6.4966', 'vk6.4987', 'vk6.4990', 'vk6.5010', 'vk6.5022', 'vk6.5243', 'vk6.5246', 'vk6.5267', 'vk6.5270', 'vk6.5728', 'vk6.5734', 'vk6.5752', 'vk6.5758', 'vk6.5759', 'vk6.5765', 'vk6.5783', 'vk6.5789', 'vk6.6040', 'vk6.6043', 'vk6.6046', 'vk6.6064', 'vk6.6067', 'vk6.6070', 'vk6.6075', 'vk6.6078', 'vk6.6099', 'vk6.6102', 'vk6.6104', 'vk6.6110', 'vk6.6128', 'vk6.6134', 'vk6.6139', 'vk6.6142', 'vk6.6163', 'vk6.6166', 'vk6.6168', 'vk6.6174', 'vk6.6192', 'vk6.6198', 'vk6.6200', 'vk6.6203', 'vk6.6206', 'vk6.6224', 'vk6.6227', 'vk6.6230', 'vk6.6280', 'vk6.6286', 'vk6.6304', 'vk6.6310', 'vk6.6440', 'vk6.6446', 'vk6.6464', 'vk6.6470', 'vk6.6499', 'vk6.6502', 'vk6.6523', 'vk6.6526', 'vk6.6559', 'vk6.6571', 'vk6.6591', 'vk6.6597', 'vk6.6615', 'vk6.6621', 'vk6.6639', 'vk6.6645', 'vk6.6663', 'vk6.6669', 'vk6.6671', 'vk6.6683', 'vk6.6719', 'vk6.6725', 'vk6.6743', 'vk6.6749', 'vk6.6859', 'vk6.6865', 'vk6.6883', 'vk6.6889', 'vk6.6903', 'vk6.6906', 'vk6.6927', 'vk6.6930', 'vk6.6935', 'vk6.6938', 'vk6.6959', 'vk6.6962', 'vk6.7063', 'vk6.7066', 'vk6.7087', 'vk6.7090', 'vk6.7095', 'vk6.7098', 'vk6.7119', 'vk6.7122', 'vk6.7159', 'vk6.7162', 'vk6.7179', 'vk6.7182', 'vk6.7239', 'vk6.7242', 'vk6.7263', 'vk6.7266', 'vk6.7271', 'vk6.7274', 'vk6.7295', 'vk6.7298', 'vk6.7335', 'vk6.7338', 'vk6.7355', 'vk6.7358', 'vk6.7795', 'vk6.7801', 'vk6.7819', 'vk6.7825', 'vk6.7826', 'vk6.7832', 'vk6.7850', 'vk6.7856', 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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

Min(phi) over symmetries of the knot is: [-2,0,1,1,1,1,2,0,0,-1]

Flat knots (up to 7 crossings) with same phi are :['4.6', '6.1770', '6.1951', '7.43053', '7.44462']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['4.6', '4.8', '6.780', '6.804', '6.914', '6.931', '6.946', '6.960', '6.1002', '6.1016', '6.1019', '6.1051', '6.1058', '6.1078', '6.1102', '6.1115', '6.1217', '6.1294', '6.1306', '6.1317', '6.1321', '6.1324', '6.1336', '6.1377', '6.1416', '6.1420', '6.1427', '6.1429', '6.1434', '6.1436', '6.1437', '6.1439', '6.1441', '6.1444', '6.1450', '6.1451', '6.1458', '6.1459', '6.1477', '6.1482', '6.1490', '6.1503', '6.1504', '6.1511', '6.1521', '6.1547', '6.1560', '6.1561', '6.1562', '6.1597', '6.1598', '6.1600', '6.1601', '6.1608', '6.1620', '6.1622', '6.1624', '6.1634', '6.1635', '6.1637', '6.1638', '6.1713', '6.1725', '6.1758', '6.1846', '6.1933', '6.1944', '6.1949', '6.1950', '6.1951']

Outer characteristic polynomial of the knot is: t^5+15t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['4.6', '7.43053', '7.44462']

2-strand cable arrow polynomial of the knot is: -32*K1**4 - 48*K1**2*K2**2 + 128*K1**2*K2 - 84*K1**2 + 48*K1*K2*K3 - 8*K2**4 + 8*K2**2*K4 - 56*K2**2 - 12*K3**2 - 2*K4**2 + 56

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['4.6']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk4.10', 'vk4.15', 'vk4.20', 'vk4.23', 'vk4.32', 'vk4.35', 'vk4.38', 'vk4.50', 'vk4.63', 'vk4.67', 'vk6.3527', 'vk6.3530', 'vk6.3551', 'vk6.3554', 'vk6.3675', 'vk6.3678', 'vk6.3699', 'vk6.3702', 'vk6.3707', 'vk6.3710', 'vk6.3731', 'vk6.3734', 'vk6.3771', 'vk6.3774', 'vk6.3795', 'vk6.3798', 'vk6.3867', 'vk6.3870', 'vk6.3891', 'vk6.3894', 'vk6.3899', 'vk6.3902', 'vk6.3923', 'vk6.3926', 'vk6.3963', 'vk6.3966', 'vk6.3987', 'vk6.3990', 'vk6.4898', 'vk6.4910', 'vk6.4915', 'vk6.4918', 'vk6.4939', 'vk6.4942', 'vk6.4963', 'vk6.4966', 'vk6.4987', 'vk6.4990', 'vk6.5010', 'vk6.5022', 'vk6.5243', 'vk6.5246', 'vk6.5267', 'vk6.5270', 'vk6.5728', 'vk6.5734', 'vk6.5752', 'vk6.5758', 'vk6.5759', 'vk6.5765', 'vk6.5783', 'vk6.5789', 'vk6.6040', 'vk6.6043', 'vk6.6046', 'vk6.6064', 'vk6.6067', 'vk6.6070', 'vk6.6075', 'vk6.6078', 'vk6.6099', 'vk6.6102', 'vk6.6104', 'vk6.6110', 'vk6.6128', 'vk6.6134', 'vk6.6139', 'vk6.6142', 'vk6.6163', 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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	O1O2O3U1O4U3U4U2
R3 orbit	{'O1O2O3U1O4U3U4U2', 'O1O2O3U1U2O4U3U4'}
R3 orbit length	2
Gauss code of -K	O1O2O3U2U4U1O4U3
Gauss code of K*	O1O2O3U4U3U1O4U2
Gauss code of -K*	O1O2O3U2O4U3U1U4
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 -2 1 0 1],[ 2 0 2 1 1],[-1 -2 0 -1 1],[ 0 -1 1 0 1],[-1 -1 -1 -1 0]]
Primitive based matrix	[[ 0 1 1 0 -2],[-1 0 1 -1 -2],[-1 -1 0 -1 -1],[ 0 1 1 0 -1],[ 2 2 1 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,0,2,-1,1,2,1,1,1]
Phi over symmetry	[-2,0,1,1,1,1,2,0,0,-1]
Phi of -K	[-2,0,1,1,1,1,2,0,0,-1]
Phi of K*	[-1,-1,0,2,-1,0,2,0,1,1]
Phi of -K*	[-2,0,1,1,1,1,2,1,1,-1]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	3z+7
Enhanced Jones-Krushkal polynomial	3w^2z+7w
Inner characteristic polynomial	t^4+9t^2
Outer characteristic polynomial	t^5+15t^3+3t
Flat arrow polynomial	-2K1*2 + K2 + 2
2-strand cable arrow polynomial	-32K14 - 48K1*2K2*2 + 128K1*2K2 - 84K12 + 48K1K2K3 - 8K24 + 8K2*2K4 - 56K22 - 12K3*2 - 2K4**2 + 56
Genus of based matrix	1
Fillings of based matrix	[[{1, 4}, {2, 3}], [{2, 4}, {1, 3}], [{3, 4}, {1, 2}], [{3, 4}, {2}, {1}]]
If K is slice	False

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