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Flat knot 6.283

Min(phi) over symmetries of the knot is: [-4,-2,1,1,2,2,0,4,4,2,3,2,3,1,2,0,1,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.283']
Arrow polynomial of the knot is: -4*K1*K2 - 2*K1*K3 + 2*K1 + K2 + 2*K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.108', '6.157', '6.283', '6.399', '6.445', '6.510']
Outer characteristic polynomial of the knot is: t^7+75t^5+70t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.283']
2-strand cable arrow polynomial of the knot is: -416*K1**4 + 320*K1**3*K2*K3 + 96*K1**3*K3*K4 - 704*K1**3*K3 - 640*K1**2*K2**2 - 256*K1**2*K2*K4 + 2824*K1**2*K2 - 1024*K1**2*K3**2 - 32*K1**2*K3*K5 - 272*K1**2*K4**2 - 64*K1**2*K4*K6 - 3976*K1**2 - 224*K1*K2**2*K3 - 32*K1*K2**2*K5 - 640*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4208*K1*K2*K3 + 2192*K1*K3*K4 + 824*K1*K4*K5 + 152*K1*K5*K6 + 32*K1*K6*K7 - 32*K2**4 - 32*K2**3*K6 - 192*K2**2*K3**2 - 80*K2**2*K4**2 + 928*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 3152*K2**2 + 808*K2*K3*K5 + 280*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 + 40*K3**2*K6 - 2400*K3**2 - 1394*K4**2 - 580*K5**2 - 200*K6**2 - 28*K7**2 - 2*K8**2 + 3682
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.283']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.14103', 'vk6.14113', 'vk6.14310', 'vk6.14330', 'vk6.15539', 'vk6.15559', 'vk6.16025', 'vk6.16035', 'vk6.16439', 'vk6.16450', 'vk6.16456', 'vk6.22845', 'vk6.22855', 'vk6.34057', 'vk6.34114', 'vk6.34451', 'vk6.34493', 'vk6.34793', 'vk6.34814', 'vk6.34828', 'vk6.42408', 'vk6.42426', 'vk6.54076', 'vk6.54086', 'vk6.54300', 'vk6.54320', 'vk6.54668', 'vk6.54689', 'vk6.54703', 'vk6.64534', 'vk6.64540', 'vk6.64736']
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 O1O2O3O4O5U1U5O6U2U6U4U3
R3 orbit {'O1O2O3O4O5U1U5O6U2U6U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U2U6U4O6U1U5
Gauss code of K* O1O2O3O4U5U1U4U3U6O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U5U2U1U4U6
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 -4 -2 2 2 1 1],[ 4 0 2 4 3 1 1],[ 2 -2 0 3 2 0 1],[-2 -4 -3 0 0 0 0],[-2 -3 -2 0 0 0 0],[-1 -1 0 0 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 2 1 1 -2 -4],[-2 0 0 0 0 -2 -3],[-2 0 0 0 0 -3 -4],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 -1 -1],[ 2 2 3 0 1 0 -2],[ 4 3 4 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,-1,2,4,0,0,0,2,3,0,0,3,4,0,0,1,1,1,2]
Phi over symmetry [-4,-2,1,1,2,2,0,4,4,2,3,2,3,1,2,0,1,1,1,1,0]
Phi of -K [-4,-2,1,1,2,2,0,4,4,2,3,2,3,1,2,0,1,1,1,1,0]
Phi of K* [-2,-2,-1,-1,2,4,0,1,1,1,2,1,1,2,3,0,2,4,3,4,0]
Phi of -K* [-4,-2,1,1,2,2,2,1,1,3,4,0,1,2,3,0,0,0,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 2z^2+15z+23
Enhanced Jones-Krushkal polynomial 2w^3z^2-8w^3z+23w^2z+23w
Inner characteristic polynomial t^6+45t^4+17t^2
Outer characteristic polynomial t^7+75t^5+70t^3+7t
Flat arrow polynomial -4*K1*K2 - 2*K1*K3 + 2*K1 + K2 + 2*K3 + K4 + 1
2-strand cable arrow polynomial -416*K1**4 + 320*K1**3*K2*K3 + 96*K1**3*K3*K4 - 704*K1**3*K3 - 640*K1**2*K2**2 - 256*K1**2*K2*K4 + 2824*K1**2*K2 - 1024*K1**2*K3**2 - 32*K1**2*K3*K5 - 272*K1**2*K4**2 - 64*K1**2*K4*K6 - 3976*K1**2 - 224*K1*K2**2*K3 - 32*K1*K2**2*K5 - 640*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4208*K1*K2*K3 + 2192*K1*K3*K4 + 824*K1*K4*K5 + 152*K1*K5*K6 + 32*K1*K6*K7 - 32*K2**4 - 32*K2**3*K6 - 192*K2**2*K3**2 - 80*K2**2*K4**2 + 928*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 3152*K2**2 + 808*K2*K3*K5 + 280*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 + 40*K3**2*K6 - 2400*K3**2 - 1394*K4**2 - 580*K5**2 - 200*K6**2 - 28*K7**2 - 2*K8**2 + 3682
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
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