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Flat knot 6.1216

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,1,2,3,-1,2,1,0,1,1,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1216']
Arrow polynomial of the knot is: 12*K1**3 - 8*K1**2 - 8*K1*K2 - 5*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.324', '6.672', '6.953', '6.1196', '6.1215', '6.1216', '6.1699']
Outer characteristic polynomial of the knot is: t^7+37t^5+82t^3+12t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1216']
2-strand cable arrow polynomial of the knot is: -640*K1**4*K2**2 + 1152*K1**4*K2 - 1440*K1**4 + 256*K1**3*K2*K3 - 64*K1**3*K3 - 2368*K1**2*K2**4 + 4992*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 11232*K1**2*K2**2 - 480*K1**2*K2*K4 + 8336*K1**2*K2 - 96*K1**2*K3**2 - 4288*K1**2 + 2976*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 2144*K1*K2**2*K3 - 384*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 6840*K1*K2*K3 + 296*K1*K3*K4 + 16*K1*K4*K5 - 864*K2**6 + 1184*K2**4*K4 - 5360*K2**4 - 160*K2**3*K6 - 1024*K2**2*K3**2 - 384*K2**2*K4**2 + 3416*K2**2*K4 - 830*K2**2 + 280*K2*K3*K5 + 56*K2*K4*K6 - 1140*K3**2 - 472*K4**2 - 36*K5**2 - 2*K6**2 + 3414
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1216']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20118', 'vk6.20122', 'vk6.21398', 'vk6.21406', 'vk6.27208', 'vk6.27216', 'vk6.28886', 'vk6.28890', 'vk6.38618', 'vk6.38626', 'vk6.40808', 'vk6.40824', 'vk6.45494', 'vk6.45510', 'vk6.47224', 'vk6.47232', 'vk6.56931', 'vk6.56939', 'vk6.58067', 'vk6.58083', 'vk6.61483', 'vk6.61499', 'vk6.62626', 'vk6.62634', 'vk6.66641', 'vk6.66645', 'vk6.67426', 'vk6.67434', 'vk6.69277', 'vk6.69285', 'vk6.70014', 'vk6.70018']
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 O1O2O3O4U3U1U2O5O6U4U5U6
R3 orbit {'O1O2O3O4U3U1U2O5O6U4U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U1O5O6U3U4U2
Gauss code of K* O1O2O3U4U5O6O4O5U2U3U1U6
Gauss code of -K* O1O2O3U1U2O4O5O6U3U6U4U5
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 0 -1 1 0 2],[ 2 0 1 0 3 1 1],[ 0 -1 0 0 2 1 1],[ 1 0 0 0 1 1 1],[-1 -3 -2 -1 0 1 2],[ 0 -1 -1 -1 -1 0 1],[-2 -1 -1 -1 -2 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -2 -1 -1 -1 -1],[-1 2 0 1 -2 -1 -3],[ 0 1 -1 0 -1 -1 -1],[ 0 1 2 1 0 0 -1],[ 1 1 1 1 0 0 0],[ 2 1 3 1 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,2,1,1,1,1,-1,2,1,3,1,1,1,0,1,0]
Phi over symmetry [-2,-1,0,0,1,2,-1,1,1,2,3,-1,2,1,0,1,1,1,0,1,1]
Phi of -K [-2,-1,0,0,1,2,1,1,1,0,3,0,1,1,2,1,2,1,-1,1,-1]
Phi of K* [-2,-1,0,0,1,2,-1,1,1,2,3,-1,2,1,0,1,1,1,0,1,1]
Phi of -K* [-2,-1,0,0,1,2,0,1,1,3,1,0,1,1,1,1,2,1,-1,1,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial -2w^4z^2+7w^3z^2-4w^3z+26w^2z+25w
Inner characteristic polynomial t^6+27t^4+20t^2
Outer characteristic polynomial t^7+37t^5+82t^3+12t
Flat arrow polynomial 12*K1**3 - 8*K1**2 - 8*K1*K2 - 5*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -640*K1**4*K2**2 + 1152*K1**4*K2 - 1440*K1**4 + 256*K1**3*K2*K3 - 64*K1**3*K3 - 2368*K1**2*K2**4 + 4992*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 11232*K1**2*K2**2 - 480*K1**2*K2*K4 + 8336*K1**2*K2 - 96*K1**2*K3**2 - 4288*K1**2 + 2976*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 2144*K1*K2**2*K3 - 384*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 6840*K1*K2*K3 + 296*K1*K3*K4 + 16*K1*K4*K5 - 864*K2**6 + 1184*K2**4*K4 - 5360*K2**4 - 160*K2**3*K6 - 1024*K2**2*K3**2 - 384*K2**2*K4**2 + 3416*K2**2*K4 - 830*K2**2 + 280*K2*K3*K5 + 56*K2*K4*K6 - 1140*K3**2 - 472*K4**2 - 36*K5**2 - 2*K6**2 + 3414
Genus of based matrix 0
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}]]
If K is slice True
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