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Flat knot 6.569

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,0,1,2,1,1,1,1,1,0,0,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.569', '7.24820']
Arrow polynomial of the knot is: 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.140', '6.569', '6.943', '6.970', '6.1234', '6.1298', '6.1311', '6.1326', '6.1500', '6.1506', '6.1708', '6.1712', '6.1720', '6.1859']
Outer characteristic polynomial of the knot is: t^7+20t^5+38t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.569', '7.24820']
2-strand cable arrow polynomial of the knot is: 1536*K1**4*K2 - 2240*K1**4 + 512*K1**3*K2*K3 - 512*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3504*K1**2*K2**2 - 320*K1**2*K2*K4 + 2616*K1**2*K2 - 448*K1**2*K3**2 - 32*K1**2*K4**2 - 88*K1**2 + 224*K1*K2**3*K3 - 416*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 2272*K1*K2*K3 + 352*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 520*K2**4 - 32*K2**3*K6 - 80*K2**2*K3**2 - 16*K2**2*K4**2 + 480*K2**2*K4 - 542*K2**2 + 88*K2*K3*K5 + 16*K2*K4*K6 - 340*K3**2 - 118*K4**2 - 20*K5**2 - 2*K6**2 + 700
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.569']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.515', 'vk6.608', 'vk6.641', 'vk6.1015', 'vk6.1110', 'vk6.1156', 'vk6.1872', 'vk6.2294', 'vk6.2516', 'vk6.2562', 'vk6.2596', 'vk6.2798', 'vk6.2895', 'vk6.2921', 'vk6.3080', 'vk6.3203', 'vk6.4629', 'vk6.5916', 'vk6.6035', 'vk6.6556', 'vk6.8076', 'vk6.9387', 'vk6.17846', 'vk6.17863', 'vk6.19065', 'vk6.19870', 'vk6.22555', 'vk6.24363', 'vk6.25681', 'vk6.26311', 'vk6.26754', 'vk6.28576', 'vk6.29805', 'vk6.39906', 'vk6.43784', 'vk6.45048', 'vk6.46845', 'vk6.48011', 'vk6.48084', 'vk6.50665']
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 O1O2O3O4O5U4U5U3O6U2U1U6
R3 orbit {'O1O2O3O4U3O5U4U5O6U2U1U6', 'O1O2O3O4O5U4U5U3O6U2U1U6'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U6U5U4O6U3U1U2
Gauss code of K* O1O2O3U4O5O6O4U6U5U3U1U2
Gauss code of -K* O1O2O3U1O4O5O6U5U6U4U3U2
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 -1 -1 0 -1 1 2],[ 1 0 0 0 -1 1 2],[ 1 0 0 0 -1 1 1],[ 0 0 0 0 -1 1 0],[ 1 1 1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-2 -2 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 0 0 0 -1 -2],[-1 0 0 -1 -1 -1 -1],[ 0 0 1 0 -1 0 0],[ 1 0 1 1 0 1 1],[ 1 1 1 0 -1 0 0],[ 1 2 1 0 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,0,0,0,1,2,1,1,1,1,1,0,0,-1,-1,0]
Phi over symmetry [-2,-1,0,1,1,1,0,0,0,1,2,1,1,1,1,1,0,0,-1,-1,0]
Phi of -K [-1,-1,-1,0,1,2,-1,-1,0,1,3,0,1,1,1,1,1,2,0,2,1]
Phi of K* [-2,-1,0,1,1,1,1,2,1,2,3,0,1,1,1,1,1,0,0,-1,-1]
Phi of -K* [-1,-1,-1,0,1,2,-1,0,0,1,1,1,1,1,0,0,1,2,1,0,0]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial 6w^3z^2+19w^2z+15w
Inner characteristic polynomial t^6+12t^4+17t^2
Outer characteristic polynomial t^7+20t^5+38t^3+3t
Flat arrow polynomial 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
2-strand cable arrow polynomial 1536*K1**4*K2 - 2240*K1**4 + 512*K1**3*K2*K3 - 512*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3504*K1**2*K2**2 - 320*K1**2*K2*K4 + 2616*K1**2*K2 - 448*K1**2*K3**2 - 32*K1**2*K4**2 - 88*K1**2 + 224*K1*K2**3*K3 - 416*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 2272*K1*K2*K3 + 352*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 520*K2**4 - 32*K2**3*K6 - 80*K2**2*K3**2 - 16*K2**2*K4**2 + 480*K2**2*K4 - 542*K2**2 + 88*K2*K3*K5 + 16*K2*K4*K6 - 340*K3**2 - 118*K4**2 - 20*K5**2 - 2*K6**2 + 700
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice False
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