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Flat knot 6.246

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,0,3,1,3,4,1,0,1,1,0,1,1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.246']
Arrow polynomial of the knot is: 12*K1**3 + 4*K1**2*K2 - 14*K1**2 - 8*K1*K2 - 2*K1*K3 - 5*K1 + 6*K2 + K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.246']
Outer characteristic polynomial of the knot is: t^7+67t^5+51t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.246']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1664*K1**4*K2**2 + 2880*K1**4*K2 - 4240*K1**4 + 800*K1**3*K2*K3 + 32*K1**3*K3*K4 - 576*K1**3*K3 - 1472*K1**2*K2**4 + 4576*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 12720*K1**2*K2**2 - 1248*K1**2*K2*K4 + 11720*K1**2*K2 - 368*K1**2*K3**2 - 64*K1**2*K3*K5 - 64*K1**2*K4**2 - 5652*K1**2 + 2496*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 2816*K1*K2**2*K3 - 640*K1*K2**2*K5 - 416*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10336*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K3**2*K5 + 1376*K1*K3*K4 + 216*K1*K4*K5 + 16*K1*K5*K6 - 224*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 800*K2**4*K4 - 4120*K2**4 + 192*K2**3*K3*K5 + 32*K2**3*K4*K6 - 128*K2**3*K6 - 1552*K2**2*K3**2 - 608*K2**2*K4**2 + 3400*K2**2*K4 - 112*K2**2*K5**2 - 8*K2**2*K6**2 - 3642*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 920*K2*K3*K5 + 216*K2*K4*K6 + 32*K2*K5*K7 + 16*K3**2*K6 - 2340*K3**2 - 896*K4**2 - 176*K5**2 - 14*K6**2 + 5222
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.246']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13369', 'vk6.13436', 'vk6.13625', 'vk6.13749', 'vk6.14151', 'vk6.14378', 'vk6.15607', 'vk6.16075', 'vk6.16473', 'vk6.16490', 'vk6.17645', 'vk6.22884', 'vk6.22917', 'vk6.24194', 'vk6.33120', 'vk6.33153', 'vk6.33215', 'vk6.33278', 'vk6.34861', 'vk6.34894', 'vk6.36449', 'vk6.42443', 'vk6.42460', 'vk6.43547', 'vk6.53553', 'vk6.53592', 'vk6.53623', 'vk6.53685', 'vk6.54721', 'vk6.55683', 'vk6.60233', 'vk6.64576']
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 O1O2O3O4O5U4O6U1U6U5U2U3
R3 orbit {'O1O2O3O4O5U4O6U1U6U5U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U4U1U6U5O6U2
Gauss code of K* O1O2O3O4O5U1U4U5U6U3O6U2
Gauss code of -K* O1O2O3O4O5U4O6U3U6U1U2U5
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 -1 2 1],[ 4 0 3 4 0 3 1],[ 0 -3 0 1 -1 1 0],[-2 -4 -1 0 -1 1 0],[ 1 0 1 1 0 1 0],[-2 -3 -1 -1 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 2 1 0 -1 -4],[-2 0 1 0 -1 -1 -4],[-2 -1 0 0 -1 -1 -3],[-1 0 0 0 0 0 -1],[ 0 1 1 0 0 -1 -3],[ 1 1 1 0 1 0 0],[ 4 4 3 1 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,1,4,-1,0,1,1,4,0,1,1,3,0,0,1,1,3,0]
Phi over symmetry [-4,-1,0,1,2,2,0,3,1,3,4,1,0,1,1,0,1,1,0,0,-1]
Phi of -K [-4,-1,0,1,2,2,3,1,4,2,3,0,2,2,2,1,1,1,1,1,-1]
Phi of K* [-2,-2,-1,0,1,4,-1,1,1,2,3,1,1,2,2,1,2,4,0,1,3]
Phi of -K* [-4,-1,0,1,2,2,0,3,1,3,4,1,0,1,1,0,1,1,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^4-2t^2
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+41t^4+13t^2+1
Outer characteristic polynomial t^7+67t^5+51t^3+7t
Flat arrow polynomial 12*K1**3 + 4*K1**2*K2 - 14*K1**2 - 8*K1*K2 - 2*K1*K3 - 5*K1 + 6*K2 + K3 + 7
2-strand cable arrow polynomial 256*K1**4*K2**3 - 1664*K1**4*K2**2 + 2880*K1**4*K2 - 4240*K1**4 + 800*K1**3*K2*K3 + 32*K1**3*K3*K4 - 576*K1**3*K3 - 1472*K1**2*K2**4 + 4576*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 12720*K1**2*K2**2 - 1248*K1**2*K2*K4 + 11720*K1**2*K2 - 368*K1**2*K3**2 - 64*K1**2*K3*K5 - 64*K1**2*K4**2 - 5652*K1**2 + 2496*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 2816*K1*K2**2*K3 - 640*K1*K2**2*K5 - 416*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10336*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K3**2*K5 + 1376*K1*K3*K4 + 216*K1*K4*K5 + 16*K1*K5*K6 - 224*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 800*K2**4*K4 - 4120*K2**4 + 192*K2**3*K3*K5 + 32*K2**3*K4*K6 - 128*K2**3*K6 - 1552*K2**2*K3**2 - 608*K2**2*K4**2 + 3400*K2**2*K4 - 112*K2**2*K5**2 - 8*K2**2*K6**2 - 3642*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 920*K2*K3*K5 + 216*K2*K4*K6 + 32*K2*K5*K7 + 16*K3**2*K6 - 2340*K3**2 - 896*K4**2 - 176*K5**2 - 14*K6**2 + 5222
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}]]
If K is slice False
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