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

Flat knot 4.2

Min(phi) over symmetries of the knot is: [-3,0,1,2,1,3,2,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['4.2', '7.33612']
Arrow polynomial of the knot is: 4K13 - 2K1*2 - 2K1K2 - 2K1 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355']
Outer characteristic polynomial of the knot is: t^5+31t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['4.2', '6.335', '7.33612']
2-strand cable arrow polynomial of the knot is: -32K14 + 64K1*2K2*3 - 208K1*2K2*2 + 224K1*2K2 - 164K12 + 32K1K23K3 + 136K1K2K3 + 8K1K3K4 - 32K26 + 32K2*4K4 - 88K24 - 32K2*2K3*2 - 8K2*2K4*2 + 40K2*2K4 - 48K22 + 8K2K3K5 - 36K32 - 14K4**2 + 116
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['4.2']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk4.22', 'vk4.24', 'vk4.39', 'vk4.42', 'vk4.62', 'vk4.66', 'vk4.78', 'vk4.79', 'vk6.10080', 'vk6.10083', 'vk6.10088', 'vk6.10091', 'vk6.10092', 'vk6.10095', 'vk6.10100', 'vk6.10103', 'vk6.10180', 'vk6.10183', 'vk6.10196', 'vk6.10199', 'vk6.10204', 'vk6.10207', 'vk6.10220', 'vk6.10223', 'vk6.10252', 'vk6.10255', 'vk6.10268', 'vk6.10271', 'vk6.10324', 'vk6.10327', 'vk6.10340', 'vk6.10343', 'vk6.10348', 'vk6.10351', 'vk6.10364', 'vk6.10367', 'vk6.10396', 'vk6.10399', 'vk6.10412', 'vk6.10415', 'vk6.11070', 'vk6.11150', 'vk6.11542', 'vk6.11544', 'vk6.11549', 'vk6.11694', 'vk6.11697', 'vk6.11702', 'vk6.11705', 'vk6.11710', 'vk6.11713', 'vk6.11718', 'vk6.11721', 'vk6.11754', 'vk6.11764', 'vk6.11800', 'vk6.11803', 'vk6.11814', 'vk6.11817', 'vk6.11875', 'vk6.11877', 'vk6.11886', 'vk6.12046', 'vk6.12049', 'vk6.12054', 'vk6.12057', 'vk6.12234', 'vk6.12343', 'vk6.12710', 'vk6.12713', 'vk6.12718', 'vk6.12721', 'vk6.12726', 'vk6.12729', 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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: [-3,0,1,2,1,3,2,1,1,1]

Flat knots (up to 7 crossings) with same phi are :['4.2', '7.33612']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355']

Outer characteristic polynomial of the knot is: t^5+31t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['4.2', '6.335', '7.33612']

2-strand cable arrow polynomial of the knot is: -32*K1**4 + 64*K1**2*K2**3 - 208*K1**2*K2**2 + 224*K1**2*K2 - 164*K1**2 + 32*K1*K2**3*K3 + 136*K1*K2*K3 + 8*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 88*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 40*K2**2*K4 - 48*K2**2 + 8*K2*K3*K5 - 36*K3**2 - 14*K4**2 + 116

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['4.2']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk4.22', 'vk4.24', 'vk4.39', 'vk4.42', 'vk4.62', 'vk4.66', 'vk4.78', 'vk4.79', 'vk6.10080', 'vk6.10083', 'vk6.10088', 'vk6.10091', 'vk6.10092', 'vk6.10095', 'vk6.10100', 'vk6.10103', 'vk6.10180', 'vk6.10183', 'vk6.10196', 'vk6.10199', 'vk6.10204', 'vk6.10207', 'vk6.10220', 'vk6.10223', 'vk6.10252', 'vk6.10255', 'vk6.10268', 'vk6.10271', 'vk6.10324', 'vk6.10327', 'vk6.10340', 'vk6.10343', 'vk6.10348', 'vk6.10351', 'vk6.10364', 'vk6.10367', 'vk6.10396', 'vk6.10399', 'vk6.10412', 'vk6.10415', 'vk6.11070', 'vk6.11150', 'vk6.11542', 'vk6.11544', 'vk6.11549', 'vk6.11694', 'vk6.11697', 'vk6.11702', 'vk6.11705', 'vk6.11710', 'vk6.11713', 'vk6.11718', 'vk6.11721', 'vk6.11754', 'vk6.11764', 'vk6.11800', 'vk6.11803', 'vk6.11814', 'vk6.11817', 'vk6.11875', 'vk6.11877', 'vk6.11886', 'vk6.12046', 'vk6.12049', 'vk6.12054', 'vk6.12057', 'vk6.12234', 'vk6.12343', 'vk6.12710', 'vk6.12713', 'vk6.12718', 'vk6.12721', 'vk6.12726', 'vk6.12729', 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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	O1O2O3O4U1U3U4U2
R3 orbit	{'O1O2O3O4U1U3U4U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U3U1U2U4
Gauss code of K*	O1O2O3O4U1U4U2U3
Gauss code of -K*	O1O2O3O4U2U3U1U4
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 -3 1 0 2],[ 3 0 3 1 2],[-1 -3 0 -1 1],[ 0 -1 1 0 1],[-2 -2 -1 -1 0]]
Primitive based matrix	[[ 0 2 1 0 -3],[-2 0 -1 -1 -2],[-1 1 0 -1 -3],[ 0 1 1 0 -1],[ 3 2 3 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,0,3,1,1,2,1,3,1]
Phi over symmetry	[-3,0,1,2,1,3,2,1,1,1]
Phi of -K	[-3,0,1,2,2,1,3,0,1,0]
Phi of K*	[-2,-1,0,3,0,1,3,0,1,2]
Phi of -K*	[-3,0,1,2,1,3,2,1,1,1]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	3z+7
Enhanced Jones-Krushkal polynomial	-2w^3z+5w^2z+7w
Inner characteristic polynomial	t^4+17t^2
Outer characteristic polynomial	t^5+31t^3+7t
Flat arrow polynomial	4K13 - 2K1*2 - 2K1K2 - 2K1 + K2 + 2
2-strand cable arrow polynomial	-32K14 + 64K1*2K2*3 - 208K1*2K2*2 + 224K1*2K2 - 164K12 + 32K1K23K3 + 136K1K2K3 + 8K1K3K4 - 32K26 + 32K2*4K4 - 88K24 - 32K2*2K3*2 - 8K2*2K4*2 + 40K2*2K4 - 48K22 + 8K2K3K5 - 36K32 - 14K4**2 + 116
Genus of based matrix	1
Fillings of based matrix	[[{1, 4}, {2, 3}], [{2, 4}, {1, 3}], [{3, 4}, {1, 2}], [{4}, {2, 3}, {1}]]
If K is slice	False

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