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Flat knot 6.1255

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,0,0,1,2,2,0,0,1,2,1,1,2,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1255']
Arrow polynomial of the knot is: 4*K1**3 + 8*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.521', '6.920', '6.1255', '6.1917']
Outer characteristic polynomial of the knot is: t^7+34t^5+37t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1255']
2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 1184*K1**4*K2 - 1776*K1**4 - 512*K1**3*K2**2*K3 + 1280*K1**3*K2*K3 - 864*K1**3*K3 - 192*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2912*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 - 256*K1**2*K2**2*K4**2 + 672*K1**2*K2**2*K4 - 10640*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 192*K1**2*K2*K4**2 - 2016*K1**2*K2*K4 + 9640*K1**2*K2 - 304*K1**2*K3**2 - 240*K1**2*K4**2 - 6000*K1**2 - 128*K1*K2**3*K3*K4 + 2304*K1*K2**3*K3 + 1056*K1*K2**2*K3*K4 - 2432*K1*K2**2*K3 + 384*K1*K2**2*K4*K5 - 512*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 960*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9424*K1*K2*K3 - 160*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1776*K1*K3*K4 + 400*K1*K4*K5 - 32*K2**6 - 64*K2**4*K4**2 + 512*K2**4*K4 - 2912*K2**4 + 64*K2**3*K3*K5 + 128*K2**3*K4*K6 - 96*K2**3*K6 - 1520*K2**2*K3**2 - 976*K2**2*K4**2 + 3256*K2**2*K4 - 128*K2**2*K5**2 - 48*K2**2*K6**2 - 3618*K2**2 + 736*K2*K3*K5 + 320*K2*K4*K6 + 8*K2*K5*K7 - 2324*K3**2 - 1176*K4**2 - 140*K5**2 - 38*K6**2 + 4742
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1255']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4724', 'vk6.5043', 'vk6.6249', 'vk6.6699', 'vk6.8225', 'vk6.8667', 'vk6.9607', 'vk6.9932', 'vk6.20298', 'vk6.21633', 'vk6.27590', 'vk6.29144', 'vk6.39012', 'vk6.41262', 'vk6.45776', 'vk6.47455', 'vk6.48756', 'vk6.48957', 'vk6.49553', 'vk6.49769', 'vk6.50770', 'vk6.50976', 'vk6.51247', 'vk6.51454', 'vk6.57149', 'vk6.58335', 'vk6.61771', 'vk6.62892', 'vk6.66770', 'vk6.67648', 'vk6.69414', 'vk6.70138']
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 O1O2O3O4U5U2U3O5O6U4U1U6
R3 orbit {'O1O2O3O4U5U2U3O5O6U4U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4U1O5O6U2U3U6
Gauss code of K* O1O2O3U1U4O5O6O4U6U2U3U5
Gauss code of -K* O1O2O3U1U4O5O6O4U3U5U6U2
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 -1 -1 1 1 -2 2],[ 1 0 -1 1 2 -1 2],[ 1 1 0 1 1 0 1],[-1 -1 -1 0 0 -1 1],[-1 -2 -1 0 0 -1 1],[ 2 1 0 1 1 0 2],[-2 -2 -1 -1 -1 -2 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 0 -1 -1 -1],[-1 1 0 0 -1 -2 -1],[ 1 1 1 1 0 1 0],[ 1 2 1 2 -1 0 -1],[ 2 2 1 1 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,1,1,1,2,2,0,1,1,1,1,2,1,-1,0,1]
Phi over symmetry [-2,-1,-1,1,1,2,0,0,1,2,2,0,0,1,2,1,1,2,-1,0,1]
Phi of -K [-2,-1,-1,1,1,2,0,1,2,2,2,1,0,1,1,1,1,2,0,0,0]
Phi of K* [-2,-1,-1,1,1,2,0,0,1,2,2,0,0,1,2,1,1,2,-1,0,1]
Phi of -K* [-2,-1,-1,1,1,2,0,1,1,1,2,1,1,1,1,1,2,2,0,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+22t^4+17t^2+1
Outer characteristic polynomial t^7+34t^5+37t^3+8t
Flat arrow polynomial 4*K1**3 + 8*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -384*K1**4*K2**2 + 1184*K1**4*K2 - 1776*K1**4 - 512*K1**3*K2**2*K3 + 1280*K1**3*K2*K3 - 864*K1**3*K3 - 192*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2912*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 - 256*K1**2*K2**2*K4**2 + 672*K1**2*K2**2*K4 - 10640*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 192*K1**2*K2*K4**2 - 2016*K1**2*K2*K4 + 9640*K1**2*K2 - 304*K1**2*K3**2 - 240*K1**2*K4**2 - 6000*K1**2 - 128*K1*K2**3*K3*K4 + 2304*K1*K2**3*K3 + 1056*K1*K2**2*K3*K4 - 2432*K1*K2**2*K3 + 384*K1*K2**2*K4*K5 - 512*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 960*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9424*K1*K2*K3 - 160*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1776*K1*K3*K4 + 400*K1*K4*K5 - 32*K2**6 - 64*K2**4*K4**2 + 512*K2**4*K4 - 2912*K2**4 + 64*K2**3*K3*K5 + 128*K2**3*K4*K6 - 96*K2**3*K6 - 1520*K2**2*K3**2 - 976*K2**2*K4**2 + 3256*K2**2*K4 - 128*K2**2*K5**2 - 48*K2**2*K6**2 - 3618*K2**2 + 736*K2*K3*K5 + 320*K2*K4*K6 + 8*K2*K5*K7 - 2324*K3**2 - 1176*K4**2 - 140*K5**2 - 38*K6**2 + 4742
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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