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Flat knot 6.1862

Min(phi) over symmetries of the knot is: [-1,-1,-1,1,1,1,-1,-1,0,1,1,0,1,0,1,2,0,1,1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1862']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1*K2 + K1 + 3*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1231', '6.1372', '6.1722', '6.1817', '6.1862', '6.2082']
Outer characteristic polynomial of the knot is: t^7+27t^5+104t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1862']
2-strand cable arrow polynomial of the knot is: 1632*K1**4*K2 - 3168*K1**4 + 1056*K1**3*K2*K3 - 320*K1**3*K3 - 128*K1**2*K2**4 + 736*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 7424*K1**2*K2**2 - 768*K1**2*K2*K4 + 8376*K1**2*K2 - 1312*K1**2*K3**2 - 64*K1**2*K4**2 - 4264*K1**2 + 736*K1*K2**3*K3 - 2400*K1*K2**2*K3 - 768*K1*K2**2*K5 - 512*K1*K2*K3*K4 + 8224*K1*K2*K3 + 1904*K1*K3*K4 + 400*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 1328*K2**4 - 96*K2**3*K6 - 544*K2**2*K3**2 - 128*K2**2*K4**2 + 2320*K2**2*K4 - 4306*K2**2 + 808*K2*K3*K5 + 72*K2*K4*K6 - 2208*K3**2 - 872*K4**2 - 248*K5**2 - 6*K6**2 + 4150
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1862']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10522', 'vk6.10531', 'vk6.10605', 'vk6.10622', 'vk6.10794', 'vk6.10809', 'vk6.10899', 'vk6.10906', 'vk6.19021', 'vk6.19032', 'vk6.19089', 'vk6.19093', 'vk6.19134', 'vk6.19138', 'vk6.25541', 'vk6.25554', 'vk6.25638', 'vk6.25649', 'vk6.25761', 'vk6.25765', 'vk6.30203', 'vk6.30212', 'vk6.30284', 'vk6.30301', 'vk6.30413', 'vk6.30428', 'vk6.37723', 'vk6.37742', 'vk6.56510', 'vk6.56513', 'vk6.66166', 'vk6.66177']
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 O1O2O3U4U5O4U1O6O5U6U3U2
R3 orbit {'O1O2O3U4U5O4U1O6O5U6U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U1U4O5O4U3O6U5U6
Gauss code of K* O1O2O3U4U3U2O5O6U5O4U1U6
Gauss code of -K* O1O2O3U4U3O5U6O4O6U2U1U5
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 1 -1],[ 1 0 1 0 1 2 -1],[-1 -1 0 0 -2 1 -1],[-1 0 0 0 -2 1 -1],[ 1 -1 2 2 0 1 0],[-1 -2 -1 -1 -1 0 -1],[ 1 1 1 1 0 1 0]]
Primitive based matrix [[ 0 1 1 1 -1 -1 -1],[-1 0 1 0 0 -1 -2],[-1 -1 0 -1 -2 -1 -1],[-1 0 1 0 -1 -1 -2],[ 1 0 2 1 0 -1 1],[ 1 1 1 1 1 0 0],[ 1 2 1 2 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,1,1,1,-1,0,0,1,2,1,2,1,1,1,1,2,1,-1,0]
Phi over symmetry [-1,-1,-1,1,1,1,-1,-1,0,1,1,0,1,0,1,2,0,1,1,-1,0]
Phi of -K [-1,-1,-1,1,1,1,-1,0,1,1,1,-1,0,1,2,1,0,0,1,1,0]
Phi of K* [-1,-1,-1,1,1,1,-1,-1,0,1,1,0,1,0,1,2,0,1,1,-1,0]
Phi of -K* [-1,-1,-1,1,1,1,-1,0,1,2,2,-1,2,0,1,1,1,1,-1,-1,0]
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+21t^4+72t^2+1
Outer characteristic polynomial t^7+27t^5+104t^3+11t
Flat arrow polynomial 4*K1**3 - 8*K1*K2 + K1 + 3*K3 + 1
2-strand cable arrow polynomial 1632*K1**4*K2 - 3168*K1**4 + 1056*K1**3*K2*K3 - 320*K1**3*K3 - 128*K1**2*K2**4 + 736*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 7424*K1**2*K2**2 - 768*K1**2*K2*K4 + 8376*K1**2*K2 - 1312*K1**2*K3**2 - 64*K1**2*K4**2 - 4264*K1**2 + 736*K1*K2**3*K3 - 2400*K1*K2**2*K3 - 768*K1*K2**2*K5 - 512*K1*K2*K3*K4 + 8224*K1*K2*K3 + 1904*K1*K3*K4 + 400*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 1328*K2**4 - 96*K2**3*K6 - 544*K2**2*K3**2 - 128*K2**2*K4**2 + 2320*K2**2*K4 - 4306*K2**2 + 808*K2*K3*K5 + 72*K2*K4*K6 - 2208*K3**2 - 872*K4**2 - 248*K5**2 - 6*K6**2 + 4150
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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