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Flat knot 6.229

Min(phi) over symmetries of the knot is: [-4,-2,1,1,2,2,0,1,4,3,4,0,2,1,2,0,0,0,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.229']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1*K2 - 2*K1*K3 - K1 - K2 + K3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.229', '6.293']
Outer characteristic polynomial of the knot is: t^7+83t^5+64t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.229']
2-strand cable arrow polynomial of the knot is: 2016*K1**4*K2 - 3616*K1**4 + 1440*K1**3*K2*K3 - 1216*K1**3*K3 - 128*K1**2*K2**4 + 1280*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7344*K1**2*K2**2 - 1600*K1**2*K2*K4 + 7728*K1**2*K2 - 1504*K1**2*K3**2 - 128*K1**2*K4**2 - 3640*K1**2 + 832*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1472*K1*K2**2*K3 - 480*K1*K2**2*K5 - 544*K1*K2*K3*K4 + 7776*K1*K2*K3 - 128*K1*K2*K4*K5 + 2312*K1*K3*K4 + 400*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1504*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 976*K2**2*K3**2 - 384*K2**2*K4**2 + 2176*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 3366*K2**2 - 64*K2*K3**2*K4 + 1096*K2*K3*K5 + 336*K2*K4*K6 + 24*K2*K5*K7 - 2100*K3**2 - 1094*K4**2 - 332*K5**2 - 66*K6**2 + 3812
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.229']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16807', 'vk6.16814', 'vk6.16862', 'vk6.16871', 'vk6.18174', 'vk6.18186', 'vk6.18509', 'vk6.18521', 'vk6.23243', 'vk6.23250', 'vk6.24629', 'vk6.25050', 'vk6.25070', 'vk6.35237', 'vk6.35262', 'vk6.36767', 'vk6.37198', 'vk6.37220', 'vk6.42754', 'vk6.42765', 'vk6.44346', 'vk6.44358', 'vk6.54994', 'vk6.55027', 'vk6.55979', 'vk6.55983', 'vk6.59390', 'vk6.59404', 'vk6.60513', 'vk6.65644', 'vk6.68180', 'vk6.68187']
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 O1O2O3O4O5U3O6U1U6U5U4U2
R3 orbit {'O1O2O3O4O5U3O6U1U6U5U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U2U1U6U5O6U3
Gauss code of K* O1O2O3O4O5U1U5U6U4U3O6U2
Gauss code of -K* O1O2O3O4O5U4O6U3U2U6U1U5
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 -4 1 -2 2 2 1],[ 4 0 4 0 4 3 1],[-1 -4 0 -2 1 1 0],[ 2 0 2 0 2 1 0],[-2 -4 -1 -2 0 0 0],[-2 -3 -1 -1 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 2 1 1 -2 -4],[-2 0 0 0 -1 -1 -3],[-2 0 0 0 -1 -2 -4],[-1 0 0 0 0 0 -1],[-1 1 1 0 0 -2 -4],[ 2 1 2 0 2 0 0],[ 4 3 4 1 4 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,-1,2,4,0,0,1,1,3,0,1,2,4,0,0,1,2,4,0]
Phi over symmetry [-4,-2,1,1,2,2,0,1,4,3,4,0,2,1,2,0,0,0,1,1,0]
Phi of -K [-4,-2,1,1,2,2,2,1,4,2,3,1,3,2,3,0,0,0,1,1,0]
Phi of K* [-2,-2,-1,-1,2,4,0,0,1,2,2,0,1,3,3,0,1,1,3,4,2]
Phi of -K* [-4,-2,1,1,2,2,0,1,4,3,4,0,2,1,2,0,0,0,1,1,0]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+53t^4+21t^2+1
Outer characteristic polynomial t^7+83t^5+64t^3+4t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 4*K1*K2 - 2*K1*K3 - K1 - K2 + K3
2-strand cable arrow polynomial 2016*K1**4*K2 - 3616*K1**4 + 1440*K1**3*K2*K3 - 1216*K1**3*K3 - 128*K1**2*K2**4 + 1280*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7344*K1**2*K2**2 - 1600*K1**2*K2*K4 + 7728*K1**2*K2 - 1504*K1**2*K3**2 - 128*K1**2*K4**2 - 3640*K1**2 + 832*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1472*K1*K2**2*K3 - 480*K1*K2**2*K5 - 544*K1*K2*K3*K4 + 7776*K1*K2*K3 - 128*K1*K2*K4*K5 + 2312*K1*K3*K4 + 400*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1504*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 976*K2**2*K3**2 - 384*K2**2*K4**2 + 2176*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 3366*K2**2 - 64*K2*K3**2*K4 + 1096*K2*K3*K5 + 336*K2*K4*K6 + 24*K2*K5*K7 - 2100*K3**2 - 1094*K4**2 - 332*K5**2 - 66*K6**2 + 3812
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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