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Flat knot 6.535

Min(phi) over symmetries of the knot is: [-3,-2,-1,2,2,2,0,1,2,3,4,0,2,1,2,2,2,3,-1,-2,0]
Flat knots (up to 7 crossings) with same phi are :['6.535']
Arrow polynomial of the knot is: 8*K1**3 - 6*K1*K2 - 3*K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.67', '6.535', '6.1347', '6.1348', '6.1368']
Outer characteristic polynomial of the knot is: t^7+87t^5+179t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.535']
2-strand cable arrow polynomial of the knot is: -32*K1**4 - 128*K1**3*K3 - 1024*K1**2*K2**4 + 2496*K1**2*K2**3 - 6624*K1**2*K2**2 - 256*K1**2*K2*K4 + 5072*K1**2*K2 - 96*K1**2*K3**2 - 3096*K1**2 + 1792*K1*K2**3*K3 - 1824*K1*K2**2*K3 - 160*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5360*K1*K2*K3 + 416*K1*K3*K4 - 576*K2**6 + 512*K2**4*K4 - 2944*K2**4 - 32*K2**3*K6 - 864*K2**2*K3**2 - 120*K2**2*K4**2 + 2136*K2**2*K4 - 902*K2**2 + 288*K2*K3*K5 + 24*K2*K4*K6 - 1128*K3**2 - 284*K4**2 - 2*K6**2 + 2186
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.535']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.74176', 'vk6.74178', 'vk6.74180', 'vk6.74182', 'vk6.74784', 'vk6.74786', 'vk6.76328', 'vk6.76334', 'vk6.76849', 'vk6.76851', 'vk6.79208', 'vk6.79214', 'vk6.79678', 'vk6.79680', 'vk6.81056', 'vk6.81058', 'vk6.82914', 'vk6.82916', 'vk6.83428', 'vk6.83430', 'vk6.85248', 'vk6.85252', 'vk6.85299', 'vk6.85303', 'vk6.85314', 'vk6.85318', 'vk6.86566', 'vk6.86569', 'vk6.87537', 'vk6.87539', 'vk6.89240', 'vk6.89244']
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 O1O2O3O4O5U3U1U2O6U5U4U6
R3 orbit {'O1O2O3O4O5U3U1U2O6U5U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U2U1O6U4U5U3
Gauss code of K* O1O2O3U4O5O6O4U2U3U1U6U5
Gauss code of -K* O1O2O3U1O4O5O6U3U2U6U4U5
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 -3 -1 -2 2 2 2],[ 3 0 1 0 4 3 2],[ 1 -1 0 0 3 2 2],[ 2 0 0 0 2 1 2],[-2 -4 -3 -2 0 0 2],[-2 -3 -2 -1 0 0 1],[-2 -2 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 2 2 2 -1 -2 -3],[-2 0 2 0 -3 -2 -4],[-2 -2 0 -1 -2 -2 -2],[-2 0 1 0 -2 -1 -3],[ 1 3 2 2 0 0 -1],[ 2 2 2 1 0 0 0],[ 3 4 2 3 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-2,1,2,3,-2,0,3,2,4,1,2,2,2,2,1,3,0,1,0]
Phi over symmetry [-3,-2,-1,2,2,2,0,1,2,3,4,0,2,1,2,2,2,3,-1,-2,0]
Phi of -K [-3,-2,-1,2,2,2,1,1,1,2,3,1,2,3,2,0,1,1,0,-2,-1]
Phi of K* [-2,-2,-2,1,2,3,-2,-1,1,2,3,0,0,2,1,1,3,2,1,1,1]
Phi of -K* [-3,-2,-1,2,2,2,0,1,2,3,4,0,2,1,2,2,2,3,-1,-2,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 2z^2+7z+7
Enhanced Jones-Krushkal polynomial -4w^4z^2+6w^3z^2-12w^3z+19w^2z+7w
Inner characteristic polynomial t^6+61t^4+61t^2
Outer characteristic polynomial t^7+87t^5+179t^3+10t
Flat arrow polynomial 8*K1**3 - 6*K1*K2 - 3*K1 + K3 + 1
2-strand cable arrow polynomial -32*K1**4 - 128*K1**3*K3 - 1024*K1**2*K2**4 + 2496*K1**2*K2**3 - 6624*K1**2*K2**2 - 256*K1**2*K2*K4 + 5072*K1**2*K2 - 96*K1**2*K3**2 - 3096*K1**2 + 1792*K1*K2**3*K3 - 1824*K1*K2**2*K3 - 160*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5360*K1*K2*K3 + 416*K1*K3*K4 - 576*K2**6 + 512*K2**4*K4 - 2944*K2**4 - 32*K2**3*K6 - 864*K2**2*K3**2 - 120*K2**2*K4**2 + 2136*K2**2*K4 - 902*K2**2 + 288*K2*K3*K5 + 24*K2*K4*K6 - 1128*K3**2 - 284*K4**2 - 2*K6**2 + 2186
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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