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Flat knot 6.862

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,1,3,0,2,1,3,2,0,0,0,1,0,1,3,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.862']
Arrow polynomial of the knot is: 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.77', '6.242', '6.331', '6.862']
Outer characteristic polynomial of the knot is: t^7+55t^5+106t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.862']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 512*K1**4*K2**2 + 1088*K1**4*K2 - 2496*K1**4 + 576*K1**3*K2*K3 - 256*K1**3*K3 - 512*K1**2*K2**4 + 960*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 5888*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 7312*K1**2*K2 - 256*K1**2*K3**2 - 96*K1**2*K4**2 - 4024*K1**2 + 1600*K1*K2**3*K3 + 640*K1*K2**2*K3*K4 - 1408*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 6240*K1*K2*K3 + 928*K1*K3*K4 + 80*K1*K4*K5 - 64*K2**6 - 128*K2**4*K3**2 - 64*K2**4*K4**2 + 192*K2**4*K4 - 1520*K2**4 + 192*K2**3*K3*K5 + 128*K2**3*K4*K6 - 128*K2**3*K6 - 1216*K2**2*K3**2 - 496*K2**2*K4**2 + 1920*K2**2*K4 - 96*K2**2*K5**2 - 48*K2**2*K6**2 - 3112*K2**2 + 528*K2*K3*K5 + 160*K2*K4*K6 - 1664*K3**2 - 576*K4**2 - 56*K5**2 - 24*K6**2 + 3630
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.862']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10916', 'vk6.10927', 'vk6.12080', 'vk6.12097', 'vk6.14486', 'vk6.15707', 'vk6.16145', 'vk6.30514', 'vk6.30544', 'vk6.31799', 'vk6.34072', 'vk6.34173', 'vk6.34508', 'vk6.51749', 'vk6.52627', 'vk6.54148', 'vk6.54341', 'vk6.54544', 'vk6.63463', 'vk6.63474']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4U1U4O5O6U5U2U3U6
R3 orbit {'O1O2O3O4U1U4O5O6U5U2U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2U3U6O5O6U1U4
Gauss code of K* O1O2O3O4U5U2U3U6O5O6U1U4
Gauss code of -K* Same
Diagrammatic symmetry type -
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 3],[ 3 0 2 3 1 0 2],[ 1 -2 0 1 0 0 3],[-1 -3 -1 0 0 0 2],[-1 -1 0 0 0 0 0],[ 1 0 0 0 0 0 1],[-3 -2 -3 -2 0 -1 0]]
Primitive based matrix [[ 0 3 1 1 -1 -1 -3],[-3 0 0 -2 -1 -3 -2],[-1 0 0 0 0 0 -1],[-1 2 0 0 0 -1 -3],[ 1 1 0 0 0 0 0],[ 1 3 0 1 0 0 -2],[ 3 2 1 3 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,1,3,0,2,1,3,2,0,0,0,1,0,1,3,0,0,2]
Phi over symmetry [-3,-1,-1,1,1,3,0,2,1,3,2,0,0,0,1,0,1,3,0,0,2]
Phi of -K [-3,-1,-1,1,1,3,0,2,1,3,4,0,1,2,1,2,2,3,0,0,2]
Phi of K* [-3,-1,-1,1,1,3,0,2,1,3,4,0,1,2,1,2,2,3,0,0,2]
Phi of -K* [-3,-1,-1,1,1,3,0,2,1,3,2,0,0,0,1,0,1,3,0,0,2]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2+21w^2z+27w
Inner characteristic polynomial t^6+33t^4+38t^2+1
Outer characteristic polynomial t^7+55t^5+106t^3+5t
Flat arrow polynomial 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 4*K2 + 5
2-strand cable arrow polynomial -128*K1**6 - 512*K1**4*K2**2 + 1088*K1**4*K2 - 2496*K1**4 + 576*K1**3*K2*K3 - 256*K1**3*K3 - 512*K1**2*K2**4 + 960*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 5888*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 7312*K1**2*K2 - 256*K1**2*K3**2 - 96*K1**2*K4**2 - 4024*K1**2 + 1600*K1*K2**3*K3 + 640*K1*K2**2*K3*K4 - 1408*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 6240*K1*K2*K3 + 928*K1*K3*K4 + 80*K1*K4*K5 - 64*K2**6 - 128*K2**4*K3**2 - 64*K2**4*K4**2 + 192*K2**4*K4 - 1520*K2**4 + 192*K2**3*K3*K5 + 128*K2**3*K4*K6 - 128*K2**3*K6 - 1216*K2**2*K3**2 - 496*K2**2*K4**2 + 1920*K2**2*K4 - 96*K2**2*K5**2 - 48*K2**2*K6**2 - 3112*K2**2 + 528*K2*K3*K5 + 160*K2*K4*K6 - 1664*K3**2 - 576*K4**2 - 56*K5**2 - 24*K6**2 + 3630
Genus of based matrix 0
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}]]
If K is slice True
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