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Flat knot 6.678

Min(phi) over symmetries of the knot is: [-3,1,1,1,1,2,2,-1,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.678']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 10*K1*K2 + 2*K1 + 4*K2 + 4*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.678']
Outer characteristic polynomial of the knot is: t^5+24t^3+19t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.678']
2-strand cable arrow polynomial of the knot is: -640*K1**6 - 128*K1**4*K2**2 + 1920*K1**4*K2 - 4288*K1**4 + 608*K1**3*K2*K3 - 800*K1**3*K3 + 384*K1**2*K2**3 - 4048*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 8376*K1**2*K2 - 1472*K1**2*K3**2 - 128*K1**2*K3*K5 - 368*K1**2*K4**2 - 5224*K1**2 + 96*K1*K2**3*K3 - 832*K1*K2**2*K3 - 32*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 7192*K1*K2*K3 + 2728*K1*K3*K4 + 648*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 528*K2**4 - 32*K2**3*K6 - 432*K2**2*K3**2 - 168*K2**2*K4**2 + 1296*K2**2*K4 - 4832*K2**2 - 64*K2*K3**2*K4 + 624*K2*K3*K5 + 176*K2*K4*K6 - 32*K3**4 + 64*K3**2*K6 - 2888*K3**2 - 1268*K4**2 - 336*K5**2 - 56*K6**2 + 5346
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.678']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10939', 'vk6.10943', 'vk6.10970', 'vk6.10974', 'vk6.12109', 'vk6.12113', 'vk6.12140', 'vk6.12144', 'vk6.13793', 'vk6.13809', 'vk6.14236', 'vk6.14240', 'vk6.14685', 'vk6.14689', 'vk6.14864', 'vk6.14880', 'vk6.15843', 'vk6.15847', 'vk6.31811', 'vk6.31815', 'vk6.33625', 'vk6.33641', 'vk6.33656', 'vk6.33672', 'vk6.51783', 'vk6.51787', 'vk6.52650', 'vk6.52654', 'vk6.53803', 'vk6.53819', 'vk6.54238', 'vk6.54242']
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 O1O2O3O4U3O5U1U4O6U5U6U2
R3 orbit {'O1O2O3O4U3O5U1U4O6U5U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U6O5U1U4O6U2
Gauss code of K* O1O2O3U4U3U5U6O5U1O4O6U2
Gauss code of -K* O1O2O3U2O4O5U3O6U4U6U1U5
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 1 1 1],[ 3 0 3 0 2 2 1],[-1 -3 0 -1 0 0 1],[ 1 0 1 0 1 1 0],[-1 -2 0 -1 0 1 1],[-1 -2 0 -1 -1 0 1],[-1 -1 -1 0 -1 -1 0]]
Primitive based matrix [[ 0 1 1 1 -3],[-1 0 1 1 -2],[-1 -1 0 1 -2],[-1 -1 -1 0 -1],[ 3 2 2 1 0]]
If based matrix primitive False
Phi of primitive based matrix [-1,-1,-1,3,-1,-1,2,-1,2,1]
Phi over symmetry [-3,1,1,1,1,2,2,-1,-1,-1]
Phi of -K [-3,1,1,1,2,2,3,-1,-1,-1]
Phi of K* [-1,-1,-1,3,-1,-1,3,-1,2,2]
Phi of -K* [-3,1,1,1,1,2,2,-1,-1,-1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial z^2+22z+41
Enhanced Jones-Krushkal polynomial w^3z^2+22w^2z+41w
Inner characteristic polynomial t^4+12t^2+1
Outer characteristic polynomial t^5+24t^3+19t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 10*K1*K2 + 2*K1 + 4*K2 + 4*K3 + 5
2-strand cable arrow polynomial -640*K1**6 - 128*K1**4*K2**2 + 1920*K1**4*K2 - 4288*K1**4 + 608*K1**3*K2*K3 - 800*K1**3*K3 + 384*K1**2*K2**3 - 4048*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 8376*K1**2*K2 - 1472*K1**2*K3**2 - 128*K1**2*K3*K5 - 368*K1**2*K4**2 - 5224*K1**2 + 96*K1*K2**3*K3 - 832*K1*K2**2*K3 - 32*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 7192*K1*K2*K3 + 2728*K1*K3*K4 + 648*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 528*K2**4 - 32*K2**3*K6 - 432*K2**2*K3**2 - 168*K2**2*K4**2 + 1296*K2**2*K4 - 4832*K2**2 - 64*K2*K3**2*K4 + 624*K2*K3*K5 + 176*K2*K4*K6 - 32*K3**4 + 64*K3**2*K6 - 2888*K3**2 - 1268*K4**2 - 336*K5**2 - 56*K6**2 + 5346
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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