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Flat knot 6.1665

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,2,1,2,0,1,1,2,0,0,1,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1665']
Arrow polynomial of the knot is: 8*K1**3 - 16*K1**2 - 8*K1*K2 - 2*K1 + 8*K2 + 2*K3 + 9
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.249', '6.968', '6.1661', '6.1665', '6.2068']
Outer characteristic polynomial of the knot is: t^7+27t^5+36t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1665', '7.35564']
2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 1824*K1**4*K2 - 4464*K1**4 + 544*K1**3*K2*K3 - 640*K1**3*K3 + 1312*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 9328*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 992*K1**2*K2*K4 + 13232*K1**2*K2 - 528*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 7796*K1**2 + 672*K1*K2**3*K3 - 2048*K1*K2**2*K3 - 448*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 10360*K1*K2*K3 + 1704*K1*K3*K4 + 168*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 1888*K2**4 - 64*K2**3*K6 - 640*K2**2*K3**2 - 128*K2**2*K4**2 + 2584*K2**2*K4 - 6316*K2**2 + 600*K2*K3*K5 + 80*K2*K4*K6 - 2932*K3**2 - 1000*K4**2 - 152*K5**2 - 12*K6**2 + 6614
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1665']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17097', 'vk6.17339', 'vk6.20588', 'vk6.21995', 'vk6.23482', 'vk6.23821', 'vk6.28054', 'vk6.29511', 'vk6.35627', 'vk6.36071', 'vk6.39468', 'vk6.41667', 'vk6.42994', 'vk6.43305', 'vk6.46056', 'vk6.47722', 'vk6.55248', 'vk6.55499', 'vk6.57458', 'vk6.58623', 'vk6.59648', 'vk6.59996', 'vk6.62133', 'vk6.63097', 'vk6.65044', 'vk6.65242', 'vk6.66990', 'vk6.67853', 'vk6.68308', 'vk6.68457', 'vk6.69609', 'vk6.70300']
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 O1O2O3U2O4U1U5U3O6O5U4U6
R3 orbit {'O1O2O3U2O4U1U5U3O6O5U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O6O4U1U6U3O5U2
Gauss code of K* O1O2O3U4U2O5O4U1U6U3O6U5
Gauss code of -K* O1O2O3U4O5U1U5U3O6O4U2U6
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 -1 2 1 0 0],[ 2 0 0 2 2 1 1],[ 1 0 0 1 1 0 0],[-2 -2 -1 0 0 -2 0],[-1 -2 -1 0 0 -1 0],[ 0 -1 0 2 1 0 0],[ 0 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 0 -2 -1 -2],[-1 0 0 0 -1 -1 -2],[ 0 0 0 0 0 0 -1],[ 0 2 1 0 0 0 -1],[ 1 1 1 0 0 0 0],[ 2 2 2 1 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,0,2,1,2,0,1,1,2,0,0,1,0,1,0]
Phi over symmetry [-2,-1,0,0,1,2,0,0,2,1,2,0,1,1,2,0,0,1,0,1,0]
Phi of -K [-2,-1,0,0,1,2,1,1,1,1,2,1,1,1,2,0,0,0,1,2,1]
Phi of K* [-2,-1,0,0,1,2,1,0,2,2,2,0,1,1,1,0,1,1,1,1,1]
Phi of -K* [-2,-1,0,0,1,2,0,1,1,2,2,0,0,1,1,0,0,0,1,2,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+17t^4+14t^2+1
Outer characteristic polynomial t^7+27t^5+36t^3+8t
Flat arrow polynomial 8*K1**3 - 16*K1**2 - 8*K1*K2 - 2*K1 + 8*K2 + 2*K3 + 9
2-strand cable arrow polynomial -320*K1**4*K2**2 + 1824*K1**4*K2 - 4464*K1**4 + 544*K1**3*K2*K3 - 640*K1**3*K3 + 1312*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 9328*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 992*K1**2*K2*K4 + 13232*K1**2*K2 - 528*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 7796*K1**2 + 672*K1*K2**3*K3 - 2048*K1*K2**2*K3 - 448*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 10360*K1*K2*K3 + 1704*K1*K3*K4 + 168*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 1888*K2**4 - 64*K2**3*K6 - 640*K2**2*K3**2 - 128*K2**2*K4**2 + 2584*K2**2*K4 - 6316*K2**2 + 600*K2*K3*K5 + 80*K2*K4*K6 - 2932*K3**2 - 1000*K4**2 - 152*K5**2 - 12*K6**2 + 6614
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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