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Flat knot 6.1699

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,1,1,1,3,0,1,1,1,-1,0,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1699']
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+29t^5+54t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1699']
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 - 960*K1**2*K2**4 + 2912*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7616*K1**2*K2**2 - 320*K1**2*K2*K4 + 6144*K1**2*K2 - 96*K1**2*K3**2 - 2880*K1**2 + 1408*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 320*K1*K2**2*K5 + 4728*K1*K2*K3 + 208*K1*K3*K4 + 8*K1*K4*K5 - 224*K2**6 + 416*K2**4*K4 - 2704*K2**4 - 64*K2**3*K6 - 544*K2**2*K3**2 - 128*K2**2*K4**2 + 1832*K2**2*K4 - 1214*K2**2 + 200*K2*K3*K5 + 24*K2*K4*K6 - 708*K3**2 - 248*K4**2 - 20*K5**2 - 2*K6**2 + 2310
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1699']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20117', 'vk6.20121', 'vk6.21395', 'vk6.21403', 'vk6.27203', 'vk6.27211', 'vk6.28883', 'vk6.28887', 'vk6.38617', 'vk6.38625', 'vk6.40805', 'vk6.40821', 'vk6.45489', 'vk6.45505', 'vk6.47221', 'vk6.47229', 'vk6.56934', 'vk6.56942', 'vk6.58072', 'vk6.58088', 'vk6.61486', 'vk6.61502', 'vk6.62627', 'vk6.62635', 'vk6.66644', 'vk6.66648', 'vk6.67431', 'vk6.67439', 'vk6.69280', 'vk6.69288', 'vk6.70015', 'vk6.70019']
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 O1O2O3U1U2O4O5U6U3O6U4U5
R3 orbit {'O1O2O3U1U2O4O5U6U3O6U4U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O6U1U6O4O5U2U3
Gauss code of K* O1O2U1O3O4U5U6U2O5O6U3U4
Gauss code of -K* O1O2U3O4O3U1U2O5O6U4U5U6
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 0 2 -1],[ 2 0 1 2 1 1 1],[ 0 -1 0 1 1 1 -1],[-1 -2 -1 0 0 1 -1],[ 0 -1 -1 0 0 1 0],[-2 -1 -1 -1 -1 0 -2],[ 1 -1 1 1 0 2 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -1 -2 -1],[-1 1 0 0 -1 -1 -2],[ 0 1 0 0 -1 0 -1],[ 0 1 1 1 0 -1 -1],[ 1 2 1 0 1 0 -1],[ 2 1 2 1 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,1,1,2,1,0,1,1,2,1,0,1,1,1,1]
Phi over symmetry [-2,-1,0,0,1,2,0,1,1,1,3,0,1,1,1,-1,0,1,1,1,0]
Phi of -K [-2,-1,0,0,1,2,0,1,1,1,3,0,1,1,1,-1,0,1,1,1,0]
Phi of K* [-2,-1,0,0,1,2,0,1,1,1,3,0,1,1,1,1,0,1,1,1,0]
Phi of -K* [-2,-1,0,0,1,2,1,1,1,2,1,0,1,1,2,-1,0,1,1,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2+22w^2z+25w
Inner characteristic polynomial t^6+19t^4+32t^2+4
Outer characteristic polynomial t^7+29t^5+54t^3+8t
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 - 960*K1**2*K2**4 + 2912*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7616*K1**2*K2**2 - 320*K1**2*K2*K4 + 6144*K1**2*K2 - 96*K1**2*K3**2 - 2880*K1**2 + 1408*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 320*K1*K2**2*K5 + 4728*K1*K2*K3 + 208*K1*K3*K4 + 8*K1*K4*K5 - 224*K2**6 + 416*K2**4*K4 - 2704*K2**4 - 64*K2**3*K6 - 544*K2**2*K3**2 - 128*K2**2*K4**2 + 1832*K2**2*K4 - 1214*K2**2 + 200*K2*K3*K5 + 24*K2*K4*K6 - 708*K3**2 - 248*K4**2 - 20*K5**2 - 2*K6**2 + 2310
Genus of based matrix 0
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}]]
If K is slice True
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