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Flat knot 6.37

Min(phi) over symmetries of the knot is: [-5,-1,1,1,1,3,1,2,4,5,3,1,2,2,2,0,0,1,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.37']
Arrow polynomial of the knot is: -8*K1**3*K2 + 4*K1**3 + 4*K1**2*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1', '6.37']
Outer characteristic polynomial of the knot is: t^7+115t^5+110t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.37']
2-strand cable arrow polynomial of the knot is: -2048*K1**2*K2**4 - 512*K1**2*K2**3*K4 + 2528*K1**2*K2**3 - 3200*K1**2*K2**2 - 128*K1**2*K2*K4 + 2024*K1**2*K2 - 1872*K1**2 + 896*K1*K2**5*K3 + 640*K1*K2**4*K3*K4 - 1152*K1*K2**4*K3 - 128*K1*K2**4*K5 - 128*K1*K2**3*K3*K4 + 3040*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 576*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2432*K1*K2*K3 + 368*K1*K3*K4 + 24*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**6*K4**2 + 384*K2**6*K4 - 992*K2**6 + 128*K2**5*K4*K6 - 128*K2**5*K6 - 1152*K2**4*K3**2 - 832*K2**4*K4**2 + 1792*K2**4*K4 - 32*K2**4*K6**2 - 1728*K2**4 + 320*K2**3*K3*K5 + 224*K2**3*K4*K6 - 32*K2**3*K6 - 736*K2**2*K3**2 - 448*K2**2*K4**2 + 1000*K2**2*K4 - 496*K2**2 + 176*K2*K3*K5 + 128*K2*K4*K6 - 840*K3**2 - 424*K4**2 - 48*K5**2 - 24*K6**2 + 1606
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.37']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19993', 'vk6.19997', 'vk6.21261', 'vk6.21267', 'vk6.27030', 'vk6.27044', 'vk6.28743', 'vk6.28749', 'vk6.38431', 'vk6.38439', 'vk6.40612', 'vk6.40624', 'vk6.45313', 'vk6.45321', 'vk6.47087', 'vk6.47090', 'vk6.56802', 'vk6.56804', 'vk6.57932', 'vk6.57936', 'vk6.61311', 'vk6.61318', 'vk6.62490', 'vk6.62494', 'vk6.66514', 'vk6.66518', 'vk6.67297', 'vk6.67304', 'vk6.69158', 'vk6.69162', 'vk6.69911', 'vk6.69913']
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 O1O2O3O4O5O6U1U4U5U6U3U2
R3 orbit {'O1O2O3O4O5O6U1U4U5U6U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U4U1U2U3U6
Gauss code of K* O1O2O3O4O5O6U1U6U5U2U3U4
Gauss code of -K* O1O2O3O4O5O6U3U4U5U2U1U6
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 -5 1 1 -1 1 3],[ 5 0 5 4 1 2 3],[-1 -5 0 0 -2 0 2],[-1 -4 0 0 -2 0 2],[ 1 -1 2 2 0 1 2],[-1 -2 0 0 -1 0 1],[-3 -3 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 3 1 1 1 -1 -5],[-3 0 -1 -2 -2 -2 -3],[-1 1 0 0 0 -1 -2],[-1 2 0 0 0 -2 -4],[-1 2 0 0 0 -2 -5],[ 1 2 1 2 2 0 -1],[ 5 3 2 4 5 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,-1,1,5,1,2,2,2,3,0,0,1,2,0,2,4,2,5,1]
Phi over symmetry [-5,-1,1,1,1,3,1,2,4,5,3,1,2,2,2,0,0,1,0,2,2]
Phi of -K [-5,-1,1,1,1,3,3,1,2,4,5,0,0,1,2,0,0,0,0,0,1]
Phi of K* [-3,-1,-1,-1,1,5,0,0,1,2,5,0,0,0,1,0,0,2,1,4,3]
Phi of -K* [-5,-1,1,1,1,3,1,2,4,5,3,1,2,2,2,0,0,1,0,2,2]
Symmetry type of based matrix c
u-polynomial t^5-t^3-2t
Normalized Jones-Krushkal polynomial 2z^2+7z+7
Enhanced Jones-Krushkal polynomial -6w^4z^2+8w^3z^2-10w^3z+17w^2z+7w
Inner characteristic polynomial t^6+77t^4+14t^2
Outer characteristic polynomial t^7+115t^5+110t^3+6t
Flat arrow polynomial -8*K1**3*K2 + 4*K1**3 + 4*K1**2*K3 + 1
2-strand cable arrow polynomial -2048*K1**2*K2**4 - 512*K1**2*K2**3*K4 + 2528*K1**2*K2**3 - 3200*K1**2*K2**2 - 128*K1**2*K2*K4 + 2024*K1**2*K2 - 1872*K1**2 + 896*K1*K2**5*K3 + 640*K1*K2**4*K3*K4 - 1152*K1*K2**4*K3 - 128*K1*K2**4*K5 - 128*K1*K2**3*K3*K4 + 3040*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 576*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2432*K1*K2*K3 + 368*K1*K3*K4 + 24*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**6*K4**2 + 384*K2**6*K4 - 992*K2**6 + 128*K2**5*K4*K6 - 128*K2**5*K6 - 1152*K2**4*K3**2 - 832*K2**4*K4**2 + 1792*K2**4*K4 - 32*K2**4*K6**2 - 1728*K2**4 + 320*K2**3*K3*K5 + 224*K2**3*K4*K6 - 32*K2**3*K6 - 736*K2**2*K3**2 - 448*K2**2*K4**2 + 1000*K2**2*K4 - 496*K2**2 + 176*K2*K3*K5 + 128*K2*K4*K6 - 840*K3**2 - 424*K4**2 - 48*K5**2 - 24*K6**2 + 1606
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
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