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Flat knot 6.1053

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,1,2,1,1,0,0,1,0,1,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.1053']
Arrow polynomial of the knot is: -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.235', '6.379', '6.411', '6.547', '6.811', '6.818', '6.823', '6.897', '6.898', '6.1008', '6.1053', '6.1109', '6.1110', '6.1130', '6.1222', '6.1239', '6.1303', '6.1307', '6.1342', '6.1491', '6.1495', '6.1496', '6.1519', '6.1592', '6.1593', '6.1642', '6.1652', '6.1653', '6.1671', '6.1673', '6.1717']
Outer characteristic polynomial of the knot is: t^7+41t^5+119t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1053']
2-strand cable arrow polynomial of the knot is: -256*K1**6 + 768*K1**4*K2 - 5056*K1**4 + 96*K1**3*K2*K3 - 704*K1**3*K3 - 2384*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 192*K1**2*K2*K4 + 8752*K1**2*K2 - 1856*K1**2*K3**2 - 32*K1**2*K3*K5 - 240*K1**2*K4**2 - 5324*K1**2 - 448*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 6640*K1*K2*K3 + 3064*K1*K3*K4 + 296*K1*K4*K5 - 96*K2**4 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 552*K2**2*K4 - 4740*K2**2 + 240*K2*K3*K5 + 32*K2*K4*K6 - 3024*K3**2 - 1184*K4**2 - 148*K5**2 - 12*K6**2 + 5462
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1053']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4767', 'vk6.4777', 'vk6.5104', 'vk6.5114', 'vk6.6337', 'vk6.6766', 'vk6.6772', 'vk6.8294', 'vk6.8304', 'vk6.8747', 'vk6.9668', 'vk6.9674', 'vk6.9979', 'vk6.9985', 'vk6.21017', 'vk6.21029', 'vk6.22441', 'vk6.22453', 'vk6.28470', 'vk6.40245', 'vk6.40249', 'vk6.42171', 'vk6.46747', 'vk6.46751', 'vk6.48801', 'vk6.49018', 'vk6.49024', 'vk6.49834', 'vk6.49844', 'vk6.51501', 'vk6.58972', 'vk6.69808']
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 O1O2O3O4U5U6O5U3O6U1U4U2
R3 orbit {'O1O2O3O4U5U6O5U3O6U1U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U1U4O5U2O6U5U6
Gauss code of K* O1O2O3U1U3U4U2O5O6U5O4U6
Gauss code of -K* O1O2O3U4O5U6O4O6U2U5U1U3
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 -2 1 0 2 -1 0],[ 2 0 2 1 2 1 2],[-1 -2 0 0 1 -2 -1],[ 0 -1 0 0 0 0 1],[-2 -2 -1 0 0 -3 -1],[ 1 -1 2 0 3 0 0],[ 0 -2 1 -1 1 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 0 -1 -3 -2],[-1 1 0 0 -1 -2 -2],[ 0 0 0 0 1 0 -1],[ 0 1 1 -1 0 0 -2],[ 1 3 2 0 0 0 -1],[ 2 2 2 1 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,0,1,3,2,0,1,2,2,-1,0,1,0,2,1]
Phi over symmetry [-2,-1,0,0,1,2,0,0,1,1,2,1,1,0,0,1,0,1,1,2,0]
Phi of -K [-2,-1,0,0,1,2,0,0,1,1,2,1,1,0,0,1,0,1,1,2,0]
Phi of K* [-2,-1,0,0,1,2,0,1,2,0,2,0,1,0,1,-1,1,0,1,1,0]
Phi of -K* [-2,-1,0,0,1,2,1,1,2,2,2,0,0,2,3,1,0,0,1,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 21z+43
Enhanced Jones-Krushkal polynomial 21w^2z+43w
Inner characteristic polynomial t^6+31t^4+87t^2+4
Outer characteristic polynomial t^7+41t^5+119t^3+7t
Flat arrow polynomial -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -256*K1**6 + 768*K1**4*K2 - 5056*K1**4 + 96*K1**3*K2*K3 - 704*K1**3*K3 - 2384*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 192*K1**2*K2*K4 + 8752*K1**2*K2 - 1856*K1**2*K3**2 - 32*K1**2*K3*K5 - 240*K1**2*K4**2 - 5324*K1**2 - 448*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 6640*K1*K2*K3 + 3064*K1*K3*K4 + 296*K1*K4*K5 - 96*K2**4 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 552*K2**2*K4 - 4740*K2**2 + 240*K2*K3*K5 + 32*K2*K4*K6 - 3024*K3**2 - 1184*K4**2 - 148*K5**2 - 12*K6**2 + 5462
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}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {2, 3}, {1}], [{6}, {2, 5}, {1, 4}, {3}]]
If K is slice False
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