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Flat knot 6.919

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,3,2,3,0,1,1,1,1,1,2,2,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.919']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.211', '6.557', '6.676', '6.685', '6.750', '6.751', '6.856', '6.919', '6.1093', '6.1371']
Outer characteristic polynomial of the knot is: t^7+60t^5+52t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.919']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 1536*K1**4*K2**2 + 3136*K1**4*K2 - 4592*K1**4 - 384*K1**3*K2**2*K3 + 1472*K1**3*K2*K3 - 928*K1**3*K3 + 384*K1**2*K2**5 - 1344*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 3008*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 + 448*K1**2*K2**2*K4 - 10256*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 992*K1**2*K2*K4 + 9976*K1**2*K2 - 720*K1**2*K3**2 - 96*K1**2*K4**2 - 4312*K1**2 + 2272*K1*K2**3*K3 + 352*K1*K2**2*K3*K4 - 1536*K1*K2**2*K3 - 608*K1*K2**2*K5 + 192*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8192*K1*K2*K3 + 1128*K1*K3*K4 + 200*K1*K4*K5 - 288*K2**6 + 352*K2**4*K4 - 1776*K2**4 - 32*K2**3*K6 - 944*K2**2*K3**2 - 152*K2**2*K4**2 + 1352*K2**2*K4 - 3132*K2**2 - 32*K2*K3**2*K4 + 504*K2*K3*K5 + 32*K2*K4*K6 - 32*K3**4 + 16*K3**2*K6 - 1848*K3**2 - 472*K4**2 - 128*K5**2 - 4*K6**2 + 4118
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.919']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.14090', 'vk6.14091', 'vk6.14285', 'vk6.14286', 'vk6.15515', 'vk6.15516', 'vk6.16013', 'vk6.16014', 'vk6.16268', 'vk6.16275', 'vk6.16279', 'vk6.22582', 'vk6.22586', 'vk6.34050', 'vk6.34097', 'vk6.34098', 'vk6.34489', 'vk6.34536', 'vk6.34540', 'vk6.34565', 'vk6.34569', 'vk6.42266', 'vk6.54059', 'vk6.54060', 'vk6.54279', 'vk6.54507', 'vk6.54561', 'vk6.54564', 'vk6.59018', 'vk6.64513', 'vk6.64514', 'vk6.64623']
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 O1O2O3O4U3U5O6O5U1U2U4U6
R3 orbit {'O1O2O3O4U3U5O6O5U1U2U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1U3U4O6O5U6U2
Gauss code of K* O1O2O3O4U1U2U5U3O5O6U4U6
Gauss code of -K* O1O2O3O4U5U1O5O6U2U6U3U4
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 -1 -1 2 1 2],[ 3 0 1 0 3 3 2],[ 1 -1 0 0 2 1 1],[ 1 0 0 0 1 1 1],[-2 -3 -2 -1 0 -2 0],[-1 -3 -1 -1 2 0 2],[-2 -2 -1 -1 0 -2 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 0 -2 -1 -1 -2],[-2 0 0 -2 -1 -2 -3],[-1 2 2 0 -1 -1 -3],[ 1 1 1 1 0 0 0],[ 1 1 2 1 0 0 -1],[ 3 2 3 3 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,0,2,1,1,2,2,1,2,3,1,1,3,0,0,1]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,3,2,3,0,1,1,1,1,1,2,2,2,0]
Phi of -K [-3,-1,-1,1,2,2,1,2,1,2,3,0,1,1,2,1,2,2,-1,-1,0]
Phi of K* [-2,-2,-1,1,1,3,0,-1,1,2,2,-1,2,2,3,1,1,1,0,1,2]
Phi of -K* [-3,-1,-1,1,2,2,0,1,3,2,3,0,1,1,1,1,1,2,2,2,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 6z^2+26z+29
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2+26w^2z+29w
Inner characteristic polynomial t^6+40t^4+18t^2+1
Outer characteristic polynomial t^7+60t^5+52t^3+7t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -128*K1**6 - 1536*K1**4*K2**2 + 3136*K1**4*K2 - 4592*K1**4 - 384*K1**3*K2**2*K3 + 1472*K1**3*K2*K3 - 928*K1**3*K3 + 384*K1**2*K2**5 - 1344*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 3008*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 + 448*K1**2*K2**2*K4 - 10256*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 992*K1**2*K2*K4 + 9976*K1**2*K2 - 720*K1**2*K3**2 - 96*K1**2*K4**2 - 4312*K1**2 + 2272*K1*K2**3*K3 + 352*K1*K2**2*K3*K4 - 1536*K1*K2**2*K3 - 608*K1*K2**2*K5 + 192*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8192*K1*K2*K3 + 1128*K1*K3*K4 + 200*K1*K4*K5 - 288*K2**6 + 352*K2**4*K4 - 1776*K2**4 - 32*K2**3*K6 - 944*K2**2*K3**2 - 152*K2**2*K4**2 + 1352*K2**2*K4 - 3132*K2**2 - 32*K2*K3**2*K4 + 504*K2*K3*K5 + 32*K2*K4*K6 - 32*K3**4 + 16*K3**2*K6 - 1848*K3**2 - 472*K4**2 - 128*K5**2 - 4*K6**2 + 4118
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{6}, {4, 5}, {2, 3}, {1}]]
If K is slice False
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