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Flat knot 6.191

Min(phi) over symmetries of the knot is: [-4,-3,1,1,2,3,0,2,4,2,5,1,2,1,3,1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.191']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 - 4*K1 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.191']
Outer characteristic polynomial of the knot is: t^7+110t^5+160t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.191']
2-strand cable arrow polynomial of the knot is: -144*K1**4 - 384*K1**2*K2**4 + 480*K1**2*K2**3 - 1744*K1**2*K2**2 + 1344*K1**2*K2 - 16*K1**2*K3**2 - 1188*K1**2 + 992*K1*K2**3*K3 + 1896*K1*K2*K3 + 64*K1*K3*K4 - 704*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 448*K2**4*K4 - 1376*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 960*K2**2*K3**2 - 80*K2**2*K4**2 + 720*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 + 20*K2**2 + 456*K2*K3*K5 + 32*K2*K4*K6 + 8*K2*K5*K7 + 8*K3**2*K6 - 636*K3**2 - 110*K4**2 - 80*K5**2 - 12*K6**2 + 1068
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.191']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71616', 'vk6.71772', 'vk6.72191', 'vk6.72343', 'vk6.73378', 'vk6.73540', 'vk6.75290', 'vk6.75555', 'vk6.77236', 'vk6.77314', 'vk6.77561', 'vk6.77676', 'vk6.78274', 'vk6.78523', 'vk6.80084', 'vk6.80233', 'vk6.81114', 'vk6.81176', 'vk6.81195', 'vk6.81235', 'vk6.81335', 'vk6.81522', 'vk6.82008', 'vk6.82430', 'vk6.82743', 'vk6.85437', 'vk6.86342', 'vk6.86919', 'vk6.87129', 'vk6.88104', 'vk6.88665', 'vk6.88769']
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 O1O2O3O4O5U2O6U1U5U6U3U4
R3 orbit {'O1O2O3O4O5U2O6U1U5U6U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U3U6U1U5O6U4
Gauss code of K* O1O2O3O4O5U1U6U4U5U2O6U3
Gauss code of -K* O1O2O3O4O5U3O6U4U1U2U6U5
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -4 -3 1 3 1 2],[ 4 0 0 4 5 2 2],[ 3 0 0 2 3 1 1],[-1 -4 -2 0 1 -1 1],[-3 -5 -3 -1 0 -1 1],[-1 -2 -1 1 1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 2 1 1 -3 -4],[-3 0 1 -1 -1 -3 -5],[-2 -1 0 -1 -1 -1 -2],[-1 1 1 0 1 -1 -2],[-1 1 1 -1 0 -2 -4],[ 3 3 1 1 2 0 0],[ 4 5 2 2 4 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,-1,3,4,-1,1,1,3,5,1,1,1,2,-1,1,2,2,4,0]
Phi over symmetry [-4,-3,1,1,2,3,0,2,4,2,5,1,2,1,3,1,1,1,1,1,-1]
Phi of -K [-4,-3,1,1,2,3,1,1,3,4,2,2,3,4,3,1,0,1,0,1,2]
Phi of K* [-3,-2,-1,-1,3,4,2,1,1,3,2,0,0,4,4,-1,2,1,3,3,1]
Phi of -K* [-4,-3,1,1,2,3,0,2,4,2,5,1,2,1,3,1,1,1,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial 4w^4z-12w^3z+4w^3+13w^2z+7w
Inner characteristic polynomial t^6+70t^4+23t^2
Outer characteristic polynomial t^7+110t^5+160t^3
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 - 4*K1 + K2 + 2
2-strand cable arrow polynomial -144*K1**4 - 384*K1**2*K2**4 + 480*K1**2*K2**3 - 1744*K1**2*K2**2 + 1344*K1**2*K2 - 16*K1**2*K3**2 - 1188*K1**2 + 992*K1*K2**3*K3 + 1896*K1*K2*K3 + 64*K1*K3*K4 - 704*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 448*K2**4*K4 - 1376*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 960*K2**2*K3**2 - 80*K2**2*K4**2 + 720*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 + 20*K2**2 + 456*K2*K3*K5 + 32*K2*K4*K6 + 8*K2*K5*K7 + 8*K3**2*K6 - 636*K3**2 - 110*K4**2 - 80*K5**2 - 12*K6**2 + 1068
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}]]
If K is slice False
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