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Flat knot 6.1213

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,1,1,2,2,1,1,2,1,1,2,2,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1213']
Arrow polynomial of the knot is: 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 3*K2 + 2*K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1213', '6.1927']
Outer characteristic polynomial of the knot is: t^7+43t^5+56t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1213']
2-strand cable arrow polynomial of the knot is: 32*K1**4*K2 - 176*K1**4 + 160*K1**3*K2*K3 - 448*K1**3*K3 + 64*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 2256*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 544*K1**2*K2*K4 + 5840*K1**2*K2 - 560*K1**2*K3**2 - 64*K1**2*K3*K5 - 16*K1**2*K4**2 - 6852*K1**2 + 544*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 - 640*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7376*K1*K2*K3 + 2208*K1*K3*K4 + 400*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 976*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 992*K2**2*K3**2 + 32*K2**2*K4**3 - 368*K2**2*K4**2 + 2560*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 5572*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1072*K2*K3*K5 - 32*K2*K4**2*K6 + 312*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 + 104*K3**2*K6 - 3320*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1590*K4**2 - 372*K5**2 - 124*K6**2 - 16*K7**2 - 2*K8**2 + 5662
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1213']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71479', 'vk6.71500', 'vk6.71540', 'vk6.71557', 'vk6.72013', 'vk6.72026', 'vk6.72066', 'vk6.72076', 'vk6.72520', 'vk6.72528', 'vk6.72642', 'vk6.72661', 'vk6.72917', 'vk6.72955', 'vk6.73114', 'vk6.73135', 'vk6.73645', 'vk6.73681', 'vk6.73698', 'vk6.77100', 'vk6.77126', 'vk6.77154', 'vk6.77179', 'vk6.77445', 'vk6.77472', 'vk6.77942', 'vk6.77963', 'vk6.78585', 'vk6.81428', 'vk6.86906', 'vk6.87248', 'vk6.89352']
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 O1O2O3O4U2U5U4O6O5U1U3U6
R3 orbit {'O1O2O3O4U2U5U4O6O5U1U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2U4O6O5U1U6U3
Gauss code of K* O1O2O3U4U2O5O6O4U5U1U6U3
Gauss code of -K* O1O2O3U4U1O5O4O6U5U2U6U3
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 -2 -2 1 2 0 1],[ 2 0 -1 2 2 1 1],[ 2 1 0 2 1 1 1],[-1 -2 -2 0 1 -2 0],[-2 -2 -1 -1 0 -2 -1],[ 0 -1 -1 2 2 0 1],[-1 -1 -1 0 1 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -2 -2],[-2 0 -1 -1 -2 -1 -2],[-1 1 0 0 -1 -1 -1],[-1 1 0 0 -2 -2 -2],[ 0 2 1 2 0 -1 -1],[ 2 1 1 2 1 0 1],[ 2 2 1 2 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,2,2,1,1,2,1,2,0,1,1,1,2,2,2,1,1,-1]
Phi over symmetry [-2,-2,0,1,1,2,-1,1,1,2,2,1,1,2,1,1,2,2,0,1,1]
Phi of -K [-2,-2,0,1,1,2,-1,1,1,2,3,1,1,2,2,-1,0,0,0,0,0]
Phi of K* [-2,-1,-1,0,2,2,0,0,0,2,3,0,-1,1,1,0,2,2,1,1,-1]
Phi of -K* [-2,-2,0,1,1,2,-1,1,1,2,2,1,1,2,1,1,2,2,0,1,1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 5z^2+24z+29
Enhanced Jones-Krushkal polynomial 5w^3z^2-2w^3z+26w^2z+29w
Inner characteristic polynomial t^6+29t^4+13t^2+1
Outer characteristic polynomial t^7+43t^5+56t^3+10t
Flat arrow polynomial 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 3*K2 + 2*K3 + K4 + 5
2-strand cable arrow polynomial 32*K1**4*K2 - 176*K1**4 + 160*K1**3*K2*K3 - 448*K1**3*K3 + 64*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 2256*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 544*K1**2*K2*K4 + 5840*K1**2*K2 - 560*K1**2*K3**2 - 64*K1**2*K3*K5 - 16*K1**2*K4**2 - 6852*K1**2 + 544*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 - 640*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7376*K1*K2*K3 + 2208*K1*K3*K4 + 400*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 976*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 992*K2**2*K3**2 + 32*K2**2*K4**3 - 368*K2**2*K4**2 + 2560*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 5572*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1072*K2*K3*K5 - 32*K2*K4**2*K6 + 312*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 + 104*K3**2*K6 - 3320*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1590*K4**2 - 372*K5**2 - 124*K6**2 - 16*K7**2 - 2*K8**2 + 5662
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}]]
If K is slice False
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