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Flat knot 6.1218

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-2,1,2,2,3,0,2,3,1,1,1,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1218']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.374', '6.446', '6.527', '6.1218', '6.1237', '6.1276', '6.1498', '6.1523', '6.1595', '6.1703', '6.1751', '6.1766', '6.1849', '6.1926']
Outer characteristic polynomial of the knot is: t^7+37t^5+76t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1218']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 448*K1**4*K2**2 + 960*K1**4*K2 - 2496*K1**4 + 512*K1**3*K2*K3 - 640*K1**3*K3 - 320*K1**2*K2**4 + 736*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 5472*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 8816*K1**2*K2 - 480*K1**2*K3**2 - 160*K1**2*K4**2 - 5912*K1**2 + 928*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1248*K1*K2**2*K3 - 32*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7336*K1*K2*K3 + 1632*K1*K3*K4 + 280*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1192*K2**4 - 1312*K2**2*K3**2 - 512*K2**2*K4**2 + 1848*K2**2*K4 - 4330*K2**2 + 696*K2*K3*K5 + 184*K2*K4*K6 - 2380*K3**2 - 994*K4**2 - 196*K5**2 - 14*K6**2 + 4976
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1218']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4691', 'vk6.4996', 'vk6.6177', 'vk6.6650', 'vk6.8174', 'vk6.8594', 'vk6.9560', 'vk6.9901', 'vk6.17395', 'vk6.20913', 'vk6.20982', 'vk6.22325', 'vk6.22404', 'vk6.23564', 'vk6.23903', 'vk6.28393', 'vk6.36163', 'vk6.40043', 'vk6.40183', 'vk6.42096', 'vk6.43076', 'vk6.43382', 'vk6.46575', 'vk6.46692', 'vk6.48723', 'vk6.49511', 'vk6.49716', 'vk6.51425', 'vk6.55553', 'vk6.58905', 'vk6.65291', 'vk6.69761']
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 O1O2O3O4U3U1U2O5O6U5U4U6
R3 orbit {'O1O2O3O4U3U1U2O5O6U5U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1U6O5O6U3U4U2
Gauss code of K* O1O2O3U4U5O4O6O5U2U3U1U6
Gauss code of -K* O1O2O3U1U3O4O5O6U2U6U4U5
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 2 -1 2],[ 2 0 1 0 3 0 1],[ 0 -1 0 0 2 0 1],[ 1 0 0 0 1 0 1],[-2 -3 -2 -1 0 0 2],[ 1 0 0 0 0 0 1],[-2 -1 -1 -1 -2 -1 0]]
Primitive based matrix [[ 0 2 2 0 -1 -1 -2],[-2 0 2 -2 0 -1 -3],[-2 -2 0 -1 -1 -1 -1],[ 0 2 1 0 0 0 -1],[ 1 0 1 0 0 0 0],[ 1 1 1 0 0 0 0],[ 2 3 1 1 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,1,1,2,-2,2,0,1,3,1,1,1,1,0,0,1,0,0,0]
Phi over symmetry [-2,-2,0,1,1,2,-2,1,2,2,3,0,2,3,1,1,1,1,0,1,1]
Phi of -K [-2,-1,-1,0,2,2,1,1,1,1,3,0,1,2,2,1,3,2,0,1,-2]
Phi of K* [-2,-2,0,1,1,2,-2,1,2,2,3,0,2,3,1,1,1,1,0,1,1]
Phi of -K* [-2,-1,-1,0,2,2,0,0,1,1,3,0,0,1,0,0,1,1,1,2,-2]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 3z^2+22z+33
Enhanced Jones-Krushkal polynomial 3w^3z^2+22w^2z+33w
Inner characteristic polynomial t^6+23t^4+23t^2+1
Outer characteristic polynomial t^7+37t^5+76t^3+6t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
2-strand cable arrow polynomial -256*K1**6 - 448*K1**4*K2**2 + 960*K1**4*K2 - 2496*K1**4 + 512*K1**3*K2*K3 - 640*K1**3*K3 - 320*K1**2*K2**4 + 736*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 5472*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 8816*K1**2*K2 - 480*K1**2*K3**2 - 160*K1**2*K4**2 - 5912*K1**2 + 928*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1248*K1*K2**2*K3 - 32*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7336*K1*K2*K3 + 1632*K1*K3*K4 + 280*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1192*K2**4 - 1312*K2**2*K3**2 - 512*K2**2*K4**2 + 1848*K2**2*K4 - 4330*K2**2 + 696*K2*K3*K5 + 184*K2*K4*K6 - 2380*K3**2 - 994*K4**2 - 196*K5**2 - 14*K6**2 + 4976
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}]]
If K is slice False
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