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Flat knot 6.294

Min(phi) over symmetries of the knot is: [-4,0,0,1,1,2,1,2,4,4,2,1,0,1,1,0,1,1,0,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.294']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 12*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 + 5*K2 + 2*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.294']
Outer characteristic polynomial of the knot is: t^7+59t^5+76t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.294']
2-strand cable arrow polynomial of the knot is: -896*K1**4*K2**2 + 1600*K1**4*K2 - 2960*K1**4 + 736*K1**3*K2*K3 - 704*K1**3*K3 - 832*K1**2*K2**4 + 2496*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 192*K1**2*K2**2*K4 - 8560*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 832*K1**2*K2*K4 + 9328*K1**2*K2 - 528*K1**2*K3**2 - 64*K1**2*K3*K5 - 5088*K1**2 + 1792*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 1888*K1*K2**2*K3 - 416*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8368*K1*K2*K3 - 32*K1*K3**2*K5 + 1176*K1*K3*K4 + 248*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 384*K2**4*K4 - 2512*K2**4 + 192*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1312*K2**2*K3**2 - 32*K2**2*K3*K7 - 544*K2**2*K4**2 + 2536*K2**2*K4 - 112*K2**2*K5**2 - 8*K2**2*K6**2 - 3648*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 952*K2*K3*K5 + 216*K2*K4*K6 + 48*K2*K5*K7 + 24*K3**2*K6 - 2156*K3**2 - 770*K4**2 - 228*K5**2 - 24*K6**2 + 4496
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.294']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13367', 'vk6.13432', 'vk6.13621', 'vk6.13747', 'vk6.14159', 'vk6.14394', 'vk6.15623', 'vk6.16083', 'vk6.16465', 'vk6.16482', 'vk6.17643', 'vk6.22868', 'vk6.22901', 'vk6.24196', 'vk6.33122', 'vk6.33157', 'vk6.33219', 'vk6.33280', 'vk6.34845', 'vk6.34878', 'vk6.36447', 'vk6.42435', 'vk6.42452', 'vk6.43549', 'vk6.53551', 'vk6.53588', 'vk6.53619', 'vk6.53683', 'vk6.54715', 'vk6.55681', 'vk6.60235', 'vk6.64582']
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 O1O2O3O4O5U1U5O6U4U6U2U3
R3 orbit {'O1O2O3O4O5U1U5O6U4U6U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U4U6U2O6U1U5
Gauss code of K* O1O2O3O4U5U3U4U1U6O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U5U4U1U2U6
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 0 2 0 1 1],[ 4 0 3 4 2 1 1],[ 0 -3 0 1 -1 0 1],[-2 -4 -1 0 -1 0 1],[ 0 -2 1 1 0 0 1],[-1 -1 0 0 0 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 2 1 1 0 0 -4],[-2 0 1 0 -1 -1 -4],[-1 -1 0 0 -1 -1 -1],[-1 0 0 0 0 0 -1],[ 0 1 1 0 0 1 -2],[ 0 1 1 0 -1 0 -3],[ 4 4 1 1 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,0,4,-1,0,1,1,4,0,1,1,1,0,0,1,-1,2,3]
Phi over symmetry [-4,0,0,1,1,2,1,2,4,4,2,1,0,1,1,0,1,1,0,2,1]
Phi of -K [-4,0,0,1,1,2,1,2,4,4,2,1,0,1,1,0,1,1,0,2,1]
Phi of K* [-2,-1,-1,0,0,4,1,2,1,1,2,0,1,1,4,0,0,4,-1,1,2]
Phi of -K* [-4,0,0,1,1,2,2,3,1,1,4,1,0,1,1,0,1,1,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 4z^2+23z+31
Enhanced Jones-Krushkal polynomial 4w^3z^2+23w^2z+31w
Inner characteristic polynomial t^6+37t^4+17t^2+1
Outer characteristic polynomial t^7+59t^5+76t^3+6t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 12*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 + 5*K2 + 2*K3 + 6
2-strand cable arrow polynomial -896*K1**4*K2**2 + 1600*K1**4*K2 - 2960*K1**4 + 736*K1**3*K2*K3 - 704*K1**3*K3 - 832*K1**2*K2**4 + 2496*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 192*K1**2*K2**2*K4 - 8560*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 832*K1**2*K2*K4 + 9328*K1**2*K2 - 528*K1**2*K3**2 - 64*K1**2*K3*K5 - 5088*K1**2 + 1792*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 1888*K1*K2**2*K3 - 416*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8368*K1*K2*K3 - 32*K1*K3**2*K5 + 1176*K1*K3*K4 + 248*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 384*K2**4*K4 - 2512*K2**4 + 192*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1312*K2**2*K3**2 - 32*K2**2*K3*K7 - 544*K2**2*K4**2 + 2536*K2**2*K4 - 112*K2**2*K5**2 - 8*K2**2*K6**2 - 3648*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 952*K2*K3*K5 + 216*K2*K4*K6 + 48*K2*K5*K7 + 24*K3**2*K6 - 2156*K3**2 - 770*K4**2 - 228*K5**2 - 24*K6**2 + 4496
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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