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Flat knot 6.1164

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,0,1,1,2,3,0,1,0,1,1,1,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1164']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.128', '6.408', '6.452', '6.532', '6.867', '6.917', '6.938', '6.1164', '6.1173', '6.1174']
Outer characteristic polynomial of the knot is: t^7+42t^5+62t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1164']
2-strand cable arrow polynomial of the knot is: 544*K1**4*K2 - 1184*K1**4 + 96*K1**3*K3*K4 - 704*K1**3*K3 + 384*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 1840*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 128*K1**2*K2*K4**2 - 1120*K1**2*K2*K4 + 5144*K1**2*K2 - 384*K1**2*K3**2 - 656*K1**2*K4**2 - 5244*K1**2 + 192*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 32*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 896*K1*K2*K3*K4 + 5360*K1*K2*K3 - 96*K1*K2*K4*K5 + 3056*K1*K3*K4 + 792*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**4*K4**2 + 224*K2**4*K4 - 696*K2**4 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 336*K2**2*K3**2 + 32*K2**2*K4**3 - 616*K2**2*K4**2 + 2368*K2**2*K4 - 8*K2**2*K6**2 - 4250*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 616*K2*K3*K5 - 32*K2*K4**2*K6 + 432*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 16*K3**2*K4**2 + 16*K3**2*K6 - 2632*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1852*K4**2 - 304*K5**2 - 86*K6**2 - 4*K7**2 - 2*K8**2 + 4540
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1164']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11198', 'vk6.11207', 'vk6.11213', 'vk6.12393', 'vk6.12395', 'vk6.12408', 'vk6.12410', 'vk6.14513', 'vk6.14526', 'vk6.15735', 'vk6.15746', 'vk6.16160', 'vk6.16165', 'vk6.30787', 'vk6.30803', 'vk6.30815', 'vk6.31991', 'vk6.31999', 'vk6.34078', 'vk6.34192', 'vk6.34475', 'vk6.34516', 'vk6.51927', 'vk6.51943', 'vk6.51965', 'vk6.54153', 'vk6.54160', 'vk6.54347', 'vk6.54360', 'vk6.63604', 'vk6.63613', 'vk6.63627']
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 O1O2O3O4U1U4U5O6O5U3U2U6
R3 orbit {'O1O2O3O4U1U4U5O6O5U3U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3U2O6O5U6U1U4
Gauss code of K* O1O2O3U4U3O5O6O4U1U6U5U2
Gauss code of -K* O1O2O3U4U1O4O5O6U5U3U2U6
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 -3 0 0 1 1 1],[ 3 0 3 2 1 3 2],[ 0 -3 0 0 0 0 1],[ 0 -2 0 0 0 0 0],[-1 -1 0 0 0 -1 0],[-1 -3 0 0 1 0 1],[-1 -2 -1 0 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 0 -3],[-1 0 1 1 0 0 -3],[-1 -1 0 0 0 0 -1],[-1 -1 0 0 0 -1 -2],[ 0 0 0 0 0 0 -2],[ 0 0 0 1 0 0 -3],[ 3 3 1 2 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,0,3,-1,-1,0,0,3,0,0,0,1,0,1,2,0,2,3]
Phi over symmetry [-3,0,0,1,1,1,0,1,1,2,3,0,1,0,1,1,1,1,-1,-1,0]
Phi of -K [-3,0,0,1,1,1,0,1,1,2,3,0,1,0,1,1,1,1,-1,-1,0]
Phi of K* [-1,-1,-1,0,0,3,-1,0,0,1,2,1,1,1,1,1,1,3,0,0,1]
Phi of -K* [-3,0,0,1,1,1,2,3,1,2,3,0,0,0,0,0,1,0,0,-1,-1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 7z^2+28z+29
Enhanced Jones-Krushkal polynomial 7w^3z^2+28w^2z+29w
Inner characteristic polynomial t^6+30t^4+24t^2+4
Outer characteristic polynomial t^7+42t^5+62t^3+13t
Flat arrow polynomial 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K3 + K4 + 2
2-strand cable arrow polynomial 544*K1**4*K2 - 1184*K1**4 + 96*K1**3*K3*K4 - 704*K1**3*K3 + 384*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 1840*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 128*K1**2*K2*K4**2 - 1120*K1**2*K2*K4 + 5144*K1**2*K2 - 384*K1**2*K3**2 - 656*K1**2*K4**2 - 5244*K1**2 + 192*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 32*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 896*K1*K2*K3*K4 + 5360*K1*K2*K3 - 96*K1*K2*K4*K5 + 3056*K1*K3*K4 + 792*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**4*K4**2 + 224*K2**4*K4 - 696*K2**4 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 336*K2**2*K3**2 + 32*K2**2*K4**3 - 616*K2**2*K4**2 + 2368*K2**2*K4 - 8*K2**2*K6**2 - 4250*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 616*K2*K3*K5 - 32*K2*K4**2*K6 + 432*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 16*K3**2*K4**2 + 16*K3**2*K6 - 2632*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1852*K4**2 - 304*K5**2 - 86*K6**2 - 4*K7**2 - 2*K8**2 + 4540
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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