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Flat knot 6.625

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,1,2,3,0,1,2,2,0,0,0,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.625']
Arrow polynomial of the knot is: -2*K1**2 - 6*K1*K2 + 3*K1 + K2 + 3*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.240', '6.577', '6.625', '6.1020']
Outer characteristic polynomial of the knot is: t^7+48t^5+68t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.625']
2-strand cable arrow polynomial of the knot is: 128*K1**4*K2 - 368*K1**4 - 832*K1**3*K3 - 336*K1**2*K2**2 - 320*K1**2*K2*K4 + 2552*K1**2*K2 - 128*K1**2*K3**2 - 32*K1**2*K3*K5 - 80*K1**2*K4**2 - 2692*K1**2 + 96*K1*K2**3*K3 - 128*K1*K2**2*K3 - 224*K1*K2*K3*K4 + 2416*K1*K2*K3 - 32*K1*K3**2*K5 + 896*K1*K3*K4 + 184*K1*K4*K5 + 8*K1*K5*K6 - 72*K2**4 - 128*K2**2*K3**2 - 24*K2**2*K4**2 + 544*K2**2*K4 - 1854*K2**2 + 248*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 24*K3**2*K6 - 1128*K3**2 - 558*K4**2 - 108*K5**2 - 18*K6**2 + 1940
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.625']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11435', 'vk6.11730', 'vk6.12749', 'vk6.13092', 'vk6.20337', 'vk6.21678', 'vk6.27641', 'vk6.29185', 'vk6.31182', 'vk6.31523', 'vk6.32350', 'vk6.32767', 'vk6.39065', 'vk6.41323', 'vk6.45821', 'vk6.47492', 'vk6.52188', 'vk6.52445', 'vk6.53019', 'vk6.53335', 'vk6.57196', 'vk6.58411', 'vk6.61810', 'vk6.62935', 'vk6.63754', 'vk6.63864', 'vk6.64182', 'vk6.64368', 'vk6.66805', 'vk6.67673', 'vk6.69445', 'vk6.70167', 'vk6.82001', 'vk6.82728', 'vk6.84320', 'vk6.85668', 'vk6.86554', 'vk6.87553', 'vk6.88277', 'vk6.89412']
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 O1O2O3O4U5O6U2O5U3U6U1U4
R3 orbit {'O1O2O3O4U5O6U2O5U3U6U1U4', 'O1O2O3O4U5U1O6O5U3U2U6U4'}
R3 orbit length 2
Gauss code of -K O1O2O3O4U1U4U5U2O6U3O5U6
Gauss code of K* O1O2O3O4U3U5U1U4O6U2O5U6
Gauss code of -K* O1O2O3O4U5O6U3O5U1U4U6U2
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 -1 -2 -1 3 0 1],[ 1 0 -1 0 3 1 1],[ 2 1 0 0 2 2 1],[ 1 0 0 0 2 1 0],[-3 -3 -2 -2 0 -2 -1],[ 0 -1 -2 -1 2 0 1],[-1 -1 -1 0 1 -1 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 -2 -2 -3 -2],[-1 1 0 -1 0 -1 -1],[ 0 2 1 0 -1 -1 -2],[ 1 2 0 1 0 0 0],[ 1 3 1 1 0 0 -1],[ 2 2 1 2 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,1,2,2,3,2,1,0,1,1,1,1,2,0,0,1]
Phi over symmetry [-3,-1,0,1,1,2,1,1,1,2,3,0,1,2,2,0,0,0,0,0,1]
Phi of -K [-2,-1,-1,0,1,3,0,1,0,2,3,0,0,1,1,0,2,2,0,1,1]
Phi of K* [-3,-1,0,1,1,2,1,1,1,2,3,0,1,2,2,0,0,0,0,0,1]
Phi of -K* [-2,-1,-1,0,1,3,0,1,2,1,2,0,1,0,2,1,1,3,1,2,1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 3z^2+16z+21
Enhanced Jones-Krushkal polynomial 3w^3z^2+16w^2z+21w
Inner characteristic polynomial t^6+32t^4+41t^2
Outer characteristic polynomial t^7+48t^5+68t^3+3t
Flat arrow polynomial -2*K1**2 - 6*K1*K2 + 3*K1 + K2 + 3*K3 + 2
2-strand cable arrow polynomial 128*K1**4*K2 - 368*K1**4 - 832*K1**3*K3 - 336*K1**2*K2**2 - 320*K1**2*K2*K4 + 2552*K1**2*K2 - 128*K1**2*K3**2 - 32*K1**2*K3*K5 - 80*K1**2*K4**2 - 2692*K1**2 + 96*K1*K2**3*K3 - 128*K1*K2**2*K3 - 224*K1*K2*K3*K4 + 2416*K1*K2*K3 - 32*K1*K3**2*K5 + 896*K1*K3*K4 + 184*K1*K4*K5 + 8*K1*K5*K6 - 72*K2**4 - 128*K2**2*K3**2 - 24*K2**2*K4**2 + 544*K2**2*K4 - 1854*K2**2 + 248*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 24*K3**2*K6 - 1128*K3**2 - 558*K4**2 - 108*K5**2 - 18*K6**2 + 1940
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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