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Flat knot 6.196

Min(phi) over symmetries of the knot is: [-4,-3,1,1,2,3,0,1,3,5,4,0,1,3,2,0,0,0,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.196']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 4*K1**2*K2 - 2*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.196']
Outer characteristic polynomial of the knot is: t^7+108t^5+182t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.196']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 224*K1**4*K2 - 1152*K1**4 + 128*K1**3*K2**3*K3 + 224*K1**3*K2*K3 + 128*K1**2*K2**5 - 1600*K1**2*K2**4 + 1088*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 2944*K1**2*K2**2 + 2904*K1**2*K2 - 288*K1**2*K3**2 - 32*K1**2*K4**2 - 1508*K1**2 + 256*K1*K2**5*K3 + 1472*K1*K2**3*K3 + 32*K1*K2*K3**3 + 2424*K1*K2*K3 + 400*K1*K3*K4 + 48*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 928*K2**6 - 384*K2**4*K3**2 - 32*K2**4*K4**2 + 448*K2**4*K4 - 712*K2**4 + 192*K2**3*K3*K5 - 720*K2**2*K3**2 - 48*K2**2*K4**2 + 400*K2**2*K4 - 32*K2**2*K5**2 - 438*K2**2 + 256*K2*K3*K5 + 8*K2*K4*K6 + 16*K3**2*K6 - 756*K3**2 - 194*K4**2 - 56*K5**2 - 10*K6**2 + 1504
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.196']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17090', 'vk6.17333', 'vk6.20246', 'vk6.21551', 'vk6.23474', 'vk6.23813', 'vk6.27469', 'vk6.29064', 'vk6.35619', 'vk6.36065', 'vk6.38880', 'vk6.41080', 'vk6.42988', 'vk6.43300', 'vk6.45645', 'vk6.47380', 'vk6.55229', 'vk6.55481', 'vk6.57079', 'vk6.58232', 'vk6.59627', 'vk6.59972', 'vk6.61617', 'vk6.62795', 'vk6.65028', 'vk6.65230', 'vk6.66709', 'vk6.67566', 'vk6.68296', 'vk6.68446', 'vk6.69359', 'vk6.70101']
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 O1O2O3O4O5U2O6U1U6U4U5U3
R3 orbit {'O1O2O3O4O5U2O6U1U6U4U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U1U2U6U5O6U4
Gauss code of K* O1O2O3O4O5U1U6U5U3U4O6U2
Gauss code of -K* O1O2O3O4O5U4O6U2U3U1U6U5
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 2 1 3 1],[ 4 0 0 5 3 4 1],[ 3 0 0 3 1 2 0],[-2 -5 -3 0 -1 1 0],[-1 -3 -1 1 0 1 0],[-3 -4 -2 -1 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 3 2 1 1 -3 -4],[-3 0 -1 0 -1 -2 -4],[-2 1 0 0 -1 -3 -5],[-1 0 0 0 0 0 -1],[-1 1 1 0 0 -1 -3],[ 3 2 3 0 1 0 0],[ 4 4 5 1 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,-1,3,4,1,0,1,2,4,0,1,3,5,0,0,1,1,3,0]
Phi over symmetry [-4,-3,1,1,2,3,0,1,3,5,4,0,1,3,2,0,0,0,1,1,1]
Phi of -K [-4,-3,1,1,2,3,1,2,4,1,3,3,4,2,4,0,0,1,1,2,0]
Phi of K* [-3,-2,-1,-1,3,4,0,1,2,4,3,0,1,2,1,0,3,2,4,4,1]
Phi of -K* [-4,-3,1,1,2,3,0,1,3,5,4,0,1,3,2,0,0,0,1,1,1]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 7z+15
Enhanced Jones-Krushkal polynomial 4w^4z-8w^3z+11w^2z+15w
Inner characteristic polynomial t^6+68t^4+45t^2
Outer characteristic polynomial t^7+108t^5+182t^3
Flat arrow polynomial -8*K1**4 + 4*K1**3 + 4*K1**2*K2 - 2*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -192*K1**4*K2**2 + 224*K1**4*K2 - 1152*K1**4 + 128*K1**3*K2**3*K3 + 224*K1**3*K2*K3 + 128*K1**2*K2**5 - 1600*K1**2*K2**4 + 1088*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 2944*K1**2*K2**2 + 2904*K1**2*K2 - 288*K1**2*K3**2 - 32*K1**2*K4**2 - 1508*K1**2 + 256*K1*K2**5*K3 + 1472*K1*K2**3*K3 + 32*K1*K2*K3**3 + 2424*K1*K2*K3 + 400*K1*K3*K4 + 48*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 928*K2**6 - 384*K2**4*K3**2 - 32*K2**4*K4**2 + 448*K2**4*K4 - 712*K2**4 + 192*K2**3*K3*K5 - 720*K2**2*K3**2 - 48*K2**2*K4**2 + 400*K2**2*K4 - 32*K2**2*K5**2 - 438*K2**2 + 256*K2*K3*K5 + 8*K2*K4*K6 + 16*K3**2*K6 - 756*K3**2 - 194*K4**2 - 56*K5**2 - 10*K6**2 + 1504
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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