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Flat knot 6.330

Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,-1,0,3,4,3,0,2,2,1,2,2,2,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.330']
Arrow polynomial of the knot is: 12*K1**3 - 4*K1**2 - 8*K1*K2 - 5*K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.330', '6.531', '6.1076', '6.1079', '6.1567']
Outer characteristic polynomial of the knot is: t^7+74t^5+82t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.330']
2-strand cable arrow polynomial of the knot is: -64*K1**4 - 64*K1**3*K3 - 192*K1**2*K2**4 + 1216*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6992*K1**2*K2**2 - 352*K1**2*K2*K4 + 7432*K1**2*K2 - 64*K1**2*K3**2 - 96*K1**2*K4**2 - 6184*K1**2 + 832*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 192*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 7848*K1*K2*K3 + 952*K1*K3*K4 + 224*K1*K4*K5 + 16*K1*K5*K6 - 224*K2**6 + 544*K2**4*K4 - 2864*K2**4 - 64*K2**3*K6 - 720*K2**2*K3**2 - 448*K2**2*K4**2 + 3376*K2**2*K4 - 4382*K2**2 - 32*K2*K3**2*K4 + 512*K2*K3*K5 + 296*K2*K4*K6 + 16*K3**2*K6 - 2392*K3**2 - 1264*K4**2 - 168*K5**2 - 82*K6**2 + 5110
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.330']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11068', 'vk6.11148', 'vk6.12232', 'vk6.12341', 'vk6.18338', 'vk6.18675', 'vk6.24774', 'vk6.25231', 'vk6.30641', 'vk6.30736', 'vk6.31871', 'vk6.31942', 'vk6.36956', 'vk6.37416', 'vk6.44145', 'vk6.44466', 'vk6.51859', 'vk6.51904', 'vk6.52722', 'vk6.52827', 'vk6.56124', 'vk6.56347', 'vk6.60641', 'vk6.60980', 'vk6.63512', 'vk6.63558', 'vk6.63992', 'vk6.64038', 'vk6.65774', 'vk6.66033', 'vk6.68777', 'vk6.68985']
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 O1O2O3O4O5U2U5O6U3U1U6U4
R3 orbit {'O1O2O3O4O5U2U5O6U3U1U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U6U5U3O6U1U4
Gauss code of K* O1O2O3O4U2U5U1U4U6O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U5U1U4U6U3
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 -3 -1 3 1 2],[ 2 0 -2 1 4 1 2],[ 3 2 0 2 3 1 1],[ 1 -1 -2 0 2 0 1],[-3 -4 -3 -2 0 0 0],[-1 -1 -1 0 0 0 0],[-2 -2 -1 -1 0 0 0]]
Primitive based matrix [[ 0 3 2 1 -1 -2 -3],[-3 0 0 0 -2 -4 -3],[-2 0 0 0 -1 -2 -1],[-1 0 0 0 0 -1 -1],[ 1 2 1 0 0 -1 -2],[ 2 4 2 1 1 0 -2],[ 3 3 1 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,2,3,0,0,2,4,3,0,1,2,1,0,1,1,1,2,2]
Phi over symmetry [-3,-2,-1,1,2,3,-1,0,3,4,3,0,2,2,1,2,2,2,1,2,1]
Phi of -K [-3,-2,-1,1,2,3,-1,0,3,4,3,0,2,2,1,2,2,2,1,2,1]
Phi of K* [-3,-2,-1,1,2,3,1,2,2,1,3,1,2,2,4,2,2,3,0,0,-1]
Phi of -K* [-3,-2,-1,1,2,3,2,2,1,1,3,1,1,2,4,0,1,2,0,0,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z^2+26z+25
Enhanced Jones-Krushkal polynomial 7w^3z^2-2w^3z+28w^2z+25w
Inner characteristic polynomial t^6+46t^4+20t^2+1
Outer characteristic polynomial t^7+74t^5+82t^3+10t
Flat arrow polynomial 12*K1**3 - 4*K1**2 - 8*K1*K2 - 5*K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -64*K1**4 - 64*K1**3*K3 - 192*K1**2*K2**4 + 1216*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6992*K1**2*K2**2 - 352*K1**2*K2*K4 + 7432*K1**2*K2 - 64*K1**2*K3**2 - 96*K1**2*K4**2 - 6184*K1**2 + 832*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 192*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 7848*K1*K2*K3 + 952*K1*K3*K4 + 224*K1*K4*K5 + 16*K1*K5*K6 - 224*K2**6 + 544*K2**4*K4 - 2864*K2**4 - 64*K2**3*K6 - 720*K2**2*K3**2 - 448*K2**2*K4**2 + 3376*K2**2*K4 - 4382*K2**2 - 32*K2*K3**2*K4 + 512*K2*K3*K5 + 296*K2*K4*K6 + 16*K3**2*K6 - 2392*K3**2 - 1264*K4**2 - 168*K5**2 - 82*K6**2 + 5110
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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