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Flat knot 6.369

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,3,1,2,0,1,1,1,2,2,2,2,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.369']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 2*K1*K3 - 2*K2**2 + 4*K2 + 2*K3 + K4 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.369']
Outer characteristic polynomial of the knot is: t^7+56t^5+49t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.369']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 768*K1**4*K2 - 2832*K1**4 + 224*K1**3*K2*K3 - 768*K1**3*K3 - 128*K1**2*K2**4 + 256*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3904*K1**2*K2**2 + 352*K1**2*K2*K3**2 + 128*K1**2*K2*K4**2 - 640*K1**2*K2*K4 + 7160*K1**2*K2 - 1232*K1**2*K3**2 - 64*K1**2*K3*K5 - 416*K1**2*K4**2 - 4472*K1**2 + 384*K1*K2**3*K3 - 1024*K1*K2**2*K3 - 160*K1*K2**2*K5 + 96*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 448*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6872*K1*K2*K3 - 192*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2320*K1*K3*K4 + 456*K1*K4*K5 + 56*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 568*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 832*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 280*K2**2*K4**2 - 32*K2**2*K4*K8 + 1136*K2**2*K4 - 8*K2**2*K6**2 - 3712*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 656*K2*K3*K5 + 256*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 96*K3**4 - 48*K3**2*K4**2 + 104*K3**2*K6 - 2264*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 884*K4**2 - 176*K5**2 - 56*K6**2 - 8*K7**2 - 2*K8**2 + 4044
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.369']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4825', 'vk6.5170', 'vk6.6387', 'vk6.6820', 'vk6.8348', 'vk6.8782', 'vk6.9718', 'vk6.10023', 'vk6.11641', 'vk6.11992', 'vk6.12983', 'vk6.20469', 'vk6.20730', 'vk6.21822', 'vk6.27853', 'vk6.29361', 'vk6.31440', 'vk6.32614', 'vk6.39283', 'vk6.39762', 'vk6.41461', 'vk6.46322', 'vk6.47584', 'vk6.47897', 'vk6.49052', 'vk6.49878', 'vk6.51304', 'vk6.51523', 'vk6.53232', 'vk6.57340', 'vk6.62026', 'vk6.64313']
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 O1O2O3O4O5U3U5O6U2U4U1U6
R3 orbit {'O1O2O3O4O5U3U5O6U2U4U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U5U2U4O6U1U3
Gauss code of K* O1O2O3O4U3U1U5U2U6O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U5U3U6U4U2
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 -2 1 1 3],[ 1 0 -1 -2 2 1 3],[ 2 1 0 -1 2 1 2],[ 2 2 1 0 2 1 1],[-1 -2 -2 -2 0 0 1],[-1 -1 -1 -1 0 0 0],[-3 -3 -2 -1 -1 0 0]]
Primitive based matrix [[ 0 3 1 1 -1 -2 -2],[-3 0 0 -1 -3 -1 -2],[-1 0 0 0 -1 -1 -1],[-1 1 0 0 -2 -2 -2],[ 1 3 1 2 0 -2 -1],[ 2 1 1 2 2 0 1],[ 2 2 1 2 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,2,2,0,1,3,1,2,0,1,1,1,2,2,2,2,1,-1]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,3,1,2,0,1,1,1,2,2,2,2,1,-1]
Phi of -K [-2,-2,-1,1,1,3,-1,-1,1,2,4,0,1,2,3,0,1,1,0,1,2]
Phi of K* [-3,-1,-1,1,2,2,1,2,1,3,4,0,0,1,1,1,2,2,0,-1,-1]
Phi of -K* [-2,-2,-1,1,1,3,-1,1,1,2,2,2,1,2,1,1,2,3,0,0,1]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2+19w^2z+31w
Inner characteristic polynomial t^6+36t^4+15t^2
Outer characteristic polynomial t^7+56t^5+49t^3+4t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 2*K1*K3 - 2*K2**2 + 4*K2 + 2*K3 + K4 + 6
2-strand cable arrow polynomial -128*K1**6 + 768*K1**4*K2 - 2832*K1**4 + 224*K1**3*K2*K3 - 768*K1**3*K3 - 128*K1**2*K2**4 + 256*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3904*K1**2*K2**2 + 352*K1**2*K2*K3**2 + 128*K1**2*K2*K4**2 - 640*K1**2*K2*K4 + 7160*K1**2*K2 - 1232*K1**2*K3**2 - 64*K1**2*K3*K5 - 416*K1**2*K4**2 - 4472*K1**2 + 384*K1*K2**3*K3 - 1024*K1*K2**2*K3 - 160*K1*K2**2*K5 + 96*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 448*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6872*K1*K2*K3 - 192*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2320*K1*K3*K4 + 456*K1*K4*K5 + 56*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 568*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 832*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 280*K2**2*K4**2 - 32*K2**2*K4*K8 + 1136*K2**2*K4 - 8*K2**2*K6**2 - 3712*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 656*K2*K3*K5 + 256*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 96*K3**4 - 48*K3**2*K4**2 + 104*K3**2*K6 - 2264*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 884*K4**2 - 176*K5**2 - 56*K6**2 - 8*K7**2 - 2*K8**2 + 4044
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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