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Flat knot 6.395

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,1,3,0,0,1,3,4,0,0,1,1,0,1,2,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.395']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.395', '6.430', '6.440', '6.548', '6.551', '6.774', '6.832', '6.887', '6.908', '6.911', '6.1205', '6.1332', '6.1339', '6.1341', '6.1346', '6.1382', '6.1488', '6.1651', '6.1655', '6.1686']
Outer characteristic polynomial of the knot is: t^7+59t^5+43t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.395']
2-strand cable arrow polynomial of the knot is: 1888*K1**4*K2 - 4544*K1**4 + 800*K1**3*K2*K3 - 1152*K1**3*K3 - 128*K1**2*K2**4 + 480*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4608*K1**2*K2**2 - 416*K1**2*K2*K4 + 7904*K1**2*K2 - 1536*K1**2*K3**2 - 160*K1**2*K3*K5 - 384*K1**2*K4**2 - 4072*K1**2 + 224*K1*K2**3*K3 - 1120*K1*K2**2*K3 - 64*K1*K2**2*K5 - 608*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6464*K1*K2*K3 - 96*K1*K2*K4*K5 + 2600*K1*K3*K4 + 904*K1*K4*K5 + 64*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 560*K2**4 - 384*K2**2*K3**2 - 112*K2**2*K4**2 + 1296*K2**2*K4 - 3950*K2**2 - 64*K2*K3**2*K4 + 784*K2*K3*K5 + 200*K2*K4*K6 + 32*K3**2*K6 - 2424*K3**2 - 1256*K4**2 - 480*K5**2 - 74*K6**2 + 4454
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.395']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10527', 'vk6.10534', 'vk6.10614', 'vk6.10629', 'vk6.10801', 'vk6.10818', 'vk6.10902', 'vk6.10911', 'vk6.19018', 'vk6.19039', 'vk6.19087', 'vk6.19099', 'vk6.19132', 'vk6.19144', 'vk6.25536', 'vk6.25555', 'vk6.25631', 'vk6.25652', 'vk6.25755', 'vk6.25767', 'vk6.30208', 'vk6.30215', 'vk6.30293', 'vk6.30308', 'vk6.30420', 'vk6.30437', 'vk6.37728', 'vk6.37741', 'vk6.56507', 'vk6.56520', 'vk6.66171', 'vk6.66176']
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 O1O2O3O4O5U4U3O6U1U6U5U2
R3 orbit {'O1O2O3O4O5U4U3O6U1U6U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U1U6U5O6U3U2
Gauss code of K* O1O2O3O4U1U4U5U6U3O6O5U2
Gauss code of -K* O1O2O3O4U3O5O6U2U6U5U1U4
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 1 -1 -1 3 1],[ 3 0 3 0 0 4 1],[-1 -3 0 -1 -1 2 0],[ 1 0 1 0 0 2 0],[ 1 0 1 0 0 1 0],[-3 -4 -2 -2 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 3 1 1 -1 -1 -3],[-3 0 0 -2 -1 -2 -4],[-1 0 0 0 0 0 -1],[-1 2 0 0 -1 -1 -3],[ 1 1 0 1 0 0 0],[ 1 2 0 1 0 0 0],[ 3 4 1 3 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,1,3,0,2,1,2,4,0,0,0,1,1,1,3,0,0,0]
Phi over symmetry [-3,-1,-1,1,1,3,0,0,1,3,4,0,0,1,1,0,1,2,0,0,2]
Phi of -K [-3,-1,-1,1,1,3,2,2,1,3,2,0,1,2,2,1,2,3,0,0,2]
Phi of K* [-3,-1,-1,1,1,3,0,2,2,3,2,0,1,1,1,2,2,3,0,2,2]
Phi of -K* [-3,-1,-1,1,1,3,0,0,1,3,4,0,0,1,1,0,1,2,0,0,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+37t^4+17t^2+1
Outer characteristic polynomial t^7+59t^5+43t^3+5t
Flat arrow polynomial 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
2-strand cable arrow polynomial 1888*K1**4*K2 - 4544*K1**4 + 800*K1**3*K2*K3 - 1152*K1**3*K3 - 128*K1**2*K2**4 + 480*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4608*K1**2*K2**2 - 416*K1**2*K2*K4 + 7904*K1**2*K2 - 1536*K1**2*K3**2 - 160*K1**2*K3*K5 - 384*K1**2*K4**2 - 4072*K1**2 + 224*K1*K2**3*K3 - 1120*K1*K2**2*K3 - 64*K1*K2**2*K5 - 608*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6464*K1*K2*K3 - 96*K1*K2*K4*K5 + 2600*K1*K3*K4 + 904*K1*K4*K5 + 64*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 560*K2**4 - 384*K2**2*K3**2 - 112*K2**2*K4**2 + 1296*K2**2*K4 - 3950*K2**2 - 64*K2*K3**2*K4 + 784*K2*K3*K5 + 200*K2*K4*K6 + 32*K3**2*K6 - 2424*K3**2 - 1256*K4**2 - 480*K5**2 - 74*K6**2 + 4454
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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