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Flat knot 6.388

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,2,3,3,0,1,2,1,0,-1,-1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.388']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 6*K1*K2 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.388', '6.687', '6.762']
Outer characteristic polynomial of the knot is: t^7+68t^5+48t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.388']
2-strand cable arrow polynomial of the knot is: -192*K1**6 - 128*K1**4*K2**2 + 1344*K1**4*K2 - 5168*K1**4 + 448*K1**3*K2*K3 - 1376*K1**3*K3 - 192*K1**2*K2**4 + 480*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 7472*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 12288*K1**2*K2 - 1392*K1**2*K3**2 - 128*K1**2*K3*K5 - 48*K1**2*K4**2 - 5876*K1**2 + 320*K1*K2**3*K3 - 864*K1*K2**2*K3 - 96*K1*K2**2*K5 + 64*K1*K2*K3**3 - 480*K1*K2*K3*K4 + 9728*K1*K2*K3 - 32*K1*K3**2*K5 + 1760*K1*K3*K4 + 288*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1616*K2**4 - 784*K2**2*K3**2 - 56*K2**2*K4**2 + 1904*K2**2*K4 - 4956*K2**2 + 704*K2*K3*K5 + 24*K2*K4*K6 - 64*K3**4 + 32*K3**2*K6 - 2576*K3**2 - 808*K4**2 - 172*K5**2 - 4*K6**2 + 5478
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.388']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11485', 'vk6.11789', 'vk6.12804', 'vk6.13140', 'vk6.17041', 'vk6.17282', 'vk6.20871', 'vk6.20948', 'vk6.22279', 'vk6.22358', 'vk6.23764', 'vk6.28341', 'vk6.31244', 'vk6.31593', 'vk6.32813', 'vk6.35552', 'vk6.36001', 'vk6.39965', 'vk6.40110', 'vk6.42039', 'vk6.42959', 'vk6.43254', 'vk6.46503', 'vk6.46628', 'vk6.52238', 'vk6.53072', 'vk6.53389', 'vk6.55448', 'vk6.58861', 'vk6.59929', 'vk6.64405', 'vk6.69725']
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 O1O2O3O4O5U4U2O6U3U5U1U6
R3 orbit {'O1O2O3O4O5U4U2O6U3U5U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U5U1U3O6U4U2
Gauss code of K* O1O2O3O4U3U5U1U6U2O6O5U4
Gauss code of -K* O1O2O3O4U1O5O6U3U6U4U5U2
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 -2 -1 -1 2 3],[ 1 0 -2 0 -1 3 3],[ 2 2 0 1 0 3 2],[ 1 0 -1 0 0 2 2],[ 1 1 0 0 0 1 1],[-2 -3 -3 -2 -1 0 1],[-3 -3 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 2 -1 -1 -1 -2],[-3 0 -1 -1 -2 -3 -2],[-2 1 0 -1 -2 -3 -3],[ 1 1 1 0 0 1 0],[ 1 2 2 0 0 0 -1],[ 1 3 3 -1 0 0 -2],[ 2 2 3 0 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,1,1,1,2,1,1,2,3,2,1,2,3,3,0,-1,0,0,1,2]
Phi over symmetry [-3,-2,1,1,1,2,0,1,2,3,3,0,1,2,1,0,-1,-1,0,0,1]
Phi of -K [-2,-1,-1,-1,2,3,-1,0,1,1,3,0,1,0,1,0,1,2,2,3,0]
Phi of K* [-3,-2,1,1,1,2,0,1,2,3,3,0,1,2,1,0,-1,-1,0,0,1]
Phi of -K* [-2,-1,-1,-1,2,3,0,1,2,3,2,0,1,1,1,0,2,2,3,3,1]
Symmetry type of based matrix c
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 3z^2+23z+35
Enhanced Jones-Krushkal polynomial 3w^3z^2+23w^2z+35w
Inner characteristic polynomial t^6+48t^4+18t^2+1
Outer characteristic polynomial t^7+68t^5+48t^3+7t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 6*K1*K2 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -192*K1**6 - 128*K1**4*K2**2 + 1344*K1**4*K2 - 5168*K1**4 + 448*K1**3*K2*K3 - 1376*K1**3*K3 - 192*K1**2*K2**4 + 480*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 7472*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 12288*K1**2*K2 - 1392*K1**2*K3**2 - 128*K1**2*K3*K5 - 48*K1**2*K4**2 - 5876*K1**2 + 320*K1*K2**3*K3 - 864*K1*K2**2*K3 - 96*K1*K2**2*K5 + 64*K1*K2*K3**3 - 480*K1*K2*K3*K4 + 9728*K1*K2*K3 - 32*K1*K3**2*K5 + 1760*K1*K3*K4 + 288*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1616*K2**4 - 784*K2**2*K3**2 - 56*K2**2*K4**2 + 1904*K2**2*K4 - 4956*K2**2 + 704*K2*K3*K5 + 24*K2*K4*K6 - 64*K3**4 + 32*K3**2*K6 - 2576*K3**2 - 808*K4**2 - 172*K5**2 - 4*K6**2 + 5478
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}]]
If K is slice False
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