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Flat knot 6.240

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,1,2,3,2,1,1,1,1,0,0,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.240']
Arrow polynomial of the knot is: -2*K1**2 - 6*K1*K2 + 3*K1 + K2 + 3*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.240', '6.577', '6.625', '6.1020']
Outer characteristic polynomial of the knot is: t^7+42t^5+46t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.240']
2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 848*K1**4 + 128*K1**3*K3*K4 - 64*K1**3*K3 - 944*K1**2*K2**2 - 992*K1**2*K2*K4 + 3208*K1**2*K2 - 384*K1**2*K3**2 - 96*K1**2*K3*K5 - 304*K1**2*K4**2 - 3868*K1**2 - 192*K1*K2**2*K3 + 32*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3696*K1*K2*K3 + 2376*K1*K3*K4 + 464*K1*K4*K5 + 8*K1*K5*K6 - 8*K2**4 - 64*K2**2*K3**2 - 56*K2**2*K4**2 + 928*K2**2*K4 - 2982*K2**2 + 272*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 2056*K3**2 - 1270*K4**2 - 212*K5**2 - 10*K6**2 + 3356
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.240']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3626', 'vk6.3709', 'vk6.3900', 'vk6.4009', 'vk6.7048', 'vk6.7097', 'vk6.7272', 'vk6.7379', 'vk6.11392', 'vk6.12575', 'vk6.12686', 'vk6.19114', 'vk6.19161', 'vk6.19820', 'vk6.25723', 'vk6.25784', 'vk6.26255', 'vk6.26698', 'vk6.30992', 'vk6.31119', 'vk6.32172', 'vk6.37842', 'vk6.37899', 'vk6.44976', 'vk6.48262', 'vk6.48441', 'vk6.50017', 'vk6.50161', 'vk6.52149', 'vk6.63723', 'vk6.66207', 'vk6.66236']
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 O1O2O3O4O5U3O6U5U6U2U1U4
R3 orbit {'O1O2O3O4O5U3O6U5U6U2U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U5U4U6U1O6U3
Gauss code of K* O1O2O3O4O5U4U3U6U5U1O6U2
Gauss code of -K* O1O2O3O4O5U4O6U5U1U6U3U2
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 -2 3 0 1],[ 1 0 0 -1 3 0 1],[ 1 0 0 -1 2 0 1],[ 2 1 1 0 2 1 1],[-3 -3 -2 -2 0 -1 1],[ 0 0 0 -1 1 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 1 -1 -2 -3 -2],[-1 -1 0 -1 -1 -1 -1],[ 0 1 1 0 0 0 -1],[ 1 2 1 0 0 0 -1],[ 1 3 1 0 0 0 -1],[ 2 2 1 1 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,-1,1,2,3,2,1,1,1,1,0,0,1,0,1,1]
Phi over symmetry [-3,-1,0,1,1,2,-1,1,2,3,2,1,1,1,1,0,0,1,0,1,1]
Phi of -K [-2,-1,-1,0,1,3,0,0,1,2,3,0,1,1,1,1,1,2,0,2,3]
Phi of K* [-3,-1,0,1,1,2,3,2,1,2,3,0,1,1,2,1,1,1,0,0,0]
Phi of -K* [-2,-1,-1,0,1,3,1,1,1,1,2,0,0,1,2,0,1,3,1,1,-1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 2z^2+15z+23
Enhanced Jones-Krushkal polynomial 2w^3z^2-8w^3z+23w^2z+23w
Inner characteristic polynomial t^6+26t^4+17t^2
Outer characteristic polynomial t^7+42t^5+46t^3+7t
Flat arrow polynomial -2*K1**2 - 6*K1*K2 + 3*K1 + K2 + 3*K3 + 2
2-strand cable arrow polynomial 96*K1**4*K2 - 848*K1**4 + 128*K1**3*K3*K4 - 64*K1**3*K3 - 944*K1**2*K2**2 - 992*K1**2*K2*K4 + 3208*K1**2*K2 - 384*K1**2*K3**2 - 96*K1**2*K3*K5 - 304*K1**2*K4**2 - 3868*K1**2 - 192*K1*K2**2*K3 + 32*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3696*K1*K2*K3 + 2376*K1*K3*K4 + 464*K1*K4*K5 + 8*K1*K5*K6 - 8*K2**4 - 64*K2**2*K3**2 - 56*K2**2*K4**2 + 928*K2**2*K4 - 2982*K2**2 + 272*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 2056*K3**2 - 1270*K4**2 - 212*K5**2 - 10*K6**2 + 3356
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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