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Flat knot 6.922

Min(phi) over symmetries of the knot is: [-3,1,1,1,1,2,3,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.922']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 10*K1*K2 + 2*K1 + 6*K2 + 4*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.383', '6.922', '6.1172', '6.1356', '6.1359']
Outer characteristic polynomial of the knot is: t^5+28t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.922']
2-strand cable arrow polynomial of the knot is: -768*K1**6 - 512*K1**4*K2**2 + 2400*K1**4*K2 - 5184*K1**4 + 1024*K1**3*K2*K3 - 864*K1**3*K3 + 384*K1**2*K2**3 - 5424*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 512*K1**2*K2*K4 + 9528*K1**2*K2 - 1664*K1**2*K3**2 - 128*K1**2*K3*K5 - 272*K1**2*K4**2 - 5352*K1**2 + 96*K1*K2**3*K3 - 928*K1*K2**2*K3 - 256*K1*K2**2*K5 - 416*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 8544*K1*K2*K3 - 32*K1*K2*K4*K5 - 64*K1*K3**2*K5 + 2928*K1*K3*K4 + 808*K1*K4*K5 + 176*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 768*K2**4 - 32*K2**3*K6 - 592*K2**2*K3**2 - 136*K2**2*K4**2 + 1704*K2**2*K4 - 5648*K2**2 - 64*K2*K3**2*K4 + 1408*K2*K3*K5 + 328*K2*K4*K6 - 32*K3**4 + 248*K3**2*K6 - 3624*K3**2 - 1592*K4**2 - 728*K5**2 - 248*K6**2 + 6326
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.922']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10951', 'vk6.10955', 'vk6.10982', 'vk6.10986', 'vk6.12121', 'vk6.12125', 'vk6.12152', 'vk6.12156', 'vk6.13789', 'vk6.13807', 'vk6.14220', 'vk6.14241', 'vk6.14669', 'vk6.14688', 'vk6.14860', 'vk6.14882', 'vk6.15827', 'vk6.15848', 'vk6.31819', 'vk6.31831', 'vk6.33629', 'vk6.33639', 'vk6.33660', 'vk6.33670', 'vk6.51779', 'vk6.51799', 'vk6.52646', 'vk6.52666', 'vk6.53799', 'vk6.53817', 'vk6.54222', 'vk6.54243']
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 O1O2O3O4U3U5O6O5U1U4U6U2
R3 orbit {'O1O2O3O4U3U5O6O5U1U4U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U1U4O6O5U6U2
Gauss code of K* O1O2O3O4U1U4U5U2O5O6U3U6
Gauss code of -K* O1O2O3O4U5U2O5O6U3U6U1U4
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 -1 1 1 1],[ 3 0 3 0 2 3 1],[-1 -3 0 -1 0 0 0],[ 1 0 1 0 1 1 1],[-1 -2 0 -1 0 -1 0],[-1 -3 0 -1 1 0 1],[-1 -1 0 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 -3],[-1 0 1 1 -3],[-1 -1 0 0 -1],[-1 -1 0 0 -2],[ 3 3 1 2 0]]
If based matrix primitive False
Phi of primitive based matrix [-1,-1,-1,3,-1,-1,3,0,1,2]
Phi over symmetry [-3,1,1,1,1,2,3,-1,-1,0]
Phi of -K [-3,1,1,1,1,2,3,-1,-1,0]
Phi of K* [-1,-1,-1,3,-1,0,2,1,1,3]
Phi of -K* [-3,1,1,1,1,2,3,0,-1,-1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial z^2+22z+41
Enhanced Jones-Krushkal polynomial w^3z^2+22w^2z+41w
Inner characteristic polynomial t^4+16t^2+1
Outer characteristic polynomial t^5+28t^3+7t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 10*K1*K2 + 2*K1 + 6*K2 + 4*K3 + 7
2-strand cable arrow polynomial -768*K1**6 - 512*K1**4*K2**2 + 2400*K1**4*K2 - 5184*K1**4 + 1024*K1**3*K2*K3 - 864*K1**3*K3 + 384*K1**2*K2**3 - 5424*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 512*K1**2*K2*K4 + 9528*K1**2*K2 - 1664*K1**2*K3**2 - 128*K1**2*K3*K5 - 272*K1**2*K4**2 - 5352*K1**2 + 96*K1*K2**3*K3 - 928*K1*K2**2*K3 - 256*K1*K2**2*K5 - 416*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 8544*K1*K2*K3 - 32*K1*K2*K4*K5 - 64*K1*K3**2*K5 + 2928*K1*K3*K4 + 808*K1*K4*K5 + 176*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 768*K2**4 - 32*K2**3*K6 - 592*K2**2*K3**2 - 136*K2**2*K4**2 + 1704*K2**2*K4 - 5648*K2**2 - 64*K2*K3**2*K4 + 1408*K2*K3*K5 + 328*K2*K4*K6 - 32*K3**4 + 248*K3**2*K6 - 3624*K3**2 - 1592*K4**2 - 728*K5**2 - 248*K6**2 + 6326
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {4, 5}, {2, 3}, {1}]]
If K is slice False
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