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Flat knot 6.1515

Min(phi) over symmetries of the knot is: [-2,0,0,2,0,1,3,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1515', '6.1704']
Arrow polynomial of the knot is: -12*K1**2 - 8*K1*K2 + 4*K1 + 6*K2 + 4*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.763', '6.1515', '6.1741', '6.1825']
Outer characteristic polynomial of the knot is: t^5+20t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1515']
2-strand cable arrow polynomial of the knot is: -640*K1**6 - 256*K1**4*K2**2 + 2176*K1**4*K2 - 5488*K1**4 + 416*K1**3*K2*K3 + 64*K1**3*K3*K4 - 384*K1**3*K3 + 32*K1**3*K4*K5 - 4048*K1**2*K2**2 - 544*K1**2*K2*K4 + 9136*K1**2*K2 - 1424*K1**2*K3**2 - 224*K1**2*K3*K5 - 480*K1**2*K4**2 - 32*K1**2*K4*K6 - 32*K1**2*K5**2 - 5308*K1**2 - 608*K1*K2**2*K3 - 32*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 6792*K1*K2*K3 - 32*K1*K3**2*K5 + 3480*K1*K3*K4 + 952*K1*K4*K5 + 48*K1*K5*K6 - 304*K2**4 - 224*K2**2*K3**2 - 64*K2**2*K4**2 + 1248*K2**2*K4 - 5136*K2**2 + 592*K2*K3*K5 + 64*K2*K4*K6 + 8*K3**2*K6 - 3140*K3**2 - 1728*K4**2 - 448*K5**2 - 24*K6**2 + 5942
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1515']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4786', 'vk6.4792', 'vk6.5121', 'vk6.5127', 'vk6.6349', 'vk6.6779', 'vk6.6789', 'vk6.8313', 'vk6.8319', 'vk6.8759', 'vk6.9683', 'vk6.9693', 'vk6.9992', 'vk6.10002', 'vk6.21007', 'vk6.21011', 'vk6.22431', 'vk6.22435', 'vk6.28459', 'vk6.40227', 'vk6.40239', 'vk6.42158', 'vk6.46725', 'vk6.46737', 'vk6.48813', 'vk6.49031', 'vk6.49041', 'vk6.49851', 'vk6.49857', 'vk6.51513', 'vk6.58959', 'vk6.69797']
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 O1O2O3U2O4U1U5O6O5U4U6U3
R3 orbit {'O1O2O3U2O4U1U5O6O5U4U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U5O6O4U6U3O5U2
Gauss code of K* O1O2O3U4U5U3O5U1O4O6U2U6
Gauss code of -K* O1O2O3U4U2O4O5U3O6U1U6U5
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 2 0 1 0],[ 2 0 0 3 1 2 1],[ 1 0 0 1 0 1 0],[-2 -3 -1 0 -1 -1 0],[ 0 -1 0 1 0 0 0],[-1 -2 -1 1 0 0 0],[ 0 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 0 0 -2],[-2 0 0 -1 -3],[ 0 0 0 0 -1],[ 0 1 0 0 -1],[ 2 3 1 1 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,0,0,2,0,1,3,0,1,1]
Phi over symmetry [-2,0,0,2,0,1,3,0,1,1]
Phi of -K [-2,0,0,2,1,1,1,0,1,2]
Phi of K* [-2,0,0,2,1,2,1,0,1,1]
Phi of -K* [-2,0,0,2,1,1,3,0,0,1]
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^4+12t^2+1
Outer characteristic polynomial t^5+20t^3+5t
Flat arrow polynomial -12*K1**2 - 8*K1*K2 + 4*K1 + 6*K2 + 4*K3 + 7
2-strand cable arrow polynomial -640*K1**6 - 256*K1**4*K2**2 + 2176*K1**4*K2 - 5488*K1**4 + 416*K1**3*K2*K3 + 64*K1**3*K3*K4 - 384*K1**3*K3 + 32*K1**3*K4*K5 - 4048*K1**2*K2**2 - 544*K1**2*K2*K4 + 9136*K1**2*K2 - 1424*K1**2*K3**2 - 224*K1**2*K3*K5 - 480*K1**2*K4**2 - 32*K1**2*K4*K6 - 32*K1**2*K5**2 - 5308*K1**2 - 608*K1*K2**2*K3 - 32*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 6792*K1*K2*K3 - 32*K1*K3**2*K5 + 3480*K1*K3*K4 + 952*K1*K4*K5 + 48*K1*K5*K6 - 304*K2**4 - 224*K2**2*K3**2 - 64*K2**2*K4**2 + 1248*K2**2*K4 - 5136*K2**2 + 592*K2*K3*K5 + 64*K2*K4*K6 + 8*K3**2*K6 - 3140*K3**2 - 1728*K4**2 - 448*K5**2 - 24*K6**2 + 5942
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}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {2, 5}, {4}, {1, 3}]]
If K is slice False
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