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Flat knot 6.890

Min(phi) over symmetries of the knot is: [-3,-2,0,0,2,3,-1,1,1,2,5,0,1,0,2,0,1,1,0,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.890']
Arrow polynomial of the knot is: 16*K1**3 - 8*K1**2 - 8*K1*K2 - 8*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.199', '6.890']
Outer characteristic polynomial of the knot is: t^7+67t^5+88t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.890']
2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 256*K1**4*K2 - 736*K1**4 + 128*K1**3*K2*K3 - 256*K1**3*K3 - 768*K1**2*K2**4 + 2624*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 7008*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 6544*K1**2*K2 - 160*K1**2*K3**2 - 4264*K1**2 + 1728*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1536*K1*K2**2*K3 - 256*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 5504*K1*K2*K3 + 480*K1*K3*K4 + 16*K1*K4*K5 + 16*K1*K5*K6 - 256*K2**6 + 192*K2**4*K4 - 2880*K2**4 - 896*K2**2*K3**2 - 128*K2**2*K4**2 + 1872*K2**2*K4 - 1448*K2**2 + 224*K2*K3*K5 + 16*K2*K4*K6 - 1160*K3**2 - 304*K4**2 - 32*K5**2 - 8*K6**2 + 2942
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.890']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70418', 'vk6.70432', 'vk6.70449', 'vk6.70458', 'vk6.70536', 'vk6.70615', 'vk6.70772', 'vk6.70853', 'vk6.70882', 'vk6.70899', 'vk6.70909', 'vk6.71019', 'vk6.71127', 'vk6.71258', 'vk6.71852', 'vk6.72300', 'vk6.76662', 'vk6.77641', 'vk6.87970', 'vk6.89208']
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 O1O2O3O4U2U3O5O6U1U5U6U4
R3 orbit {'O1O2O3O4U2U3O5O6U1U5U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U6U4O5O6U2U3
Gauss code of K* O1O2O3O4U1U5U6U4O5O6U2U3
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 -2 0 3 0 2],[ 3 0 -1 1 5 1 2],[ 2 1 0 1 2 0 0],[ 0 -1 -1 0 1 0 0],[-3 -5 -2 -1 0 -1 1],[ 0 -1 0 0 1 0 1],[-2 -2 0 0 -1 -1 0]]
Primitive based matrix [[ 0 3 2 0 0 -2 -3],[-3 0 1 -1 -1 -2 -5],[-2 -1 0 0 -1 0 -2],[ 0 1 0 0 0 -1 -1],[ 0 1 1 0 0 0 -1],[ 2 2 0 1 0 0 1],[ 3 5 2 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,0,2,3,-1,1,1,2,5,0,1,0,2,0,1,1,0,1,-1]
Phi over symmetry [-3,-2,0,0,2,3,-1,1,1,2,5,0,1,0,2,0,1,1,0,1,-1]
Phi of -K [-3,-2,0,0,2,3,2,2,2,3,1,1,2,4,3,0,2,2,1,2,2]
Phi of K* [-3,-2,0,0,2,3,2,2,2,3,1,1,2,4,3,0,2,2,1,2,2]
Phi of -K* [-3,-2,0,0,2,3,-1,1,1,2,5,0,1,0,2,0,1,1,0,1,-1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2+22w^2z+25w
Inner characteristic polynomial t^6+41t^4+50t^2+1
Outer characteristic polynomial t^7+67t^5+88t^3+5t
Flat arrow polynomial 16*K1**3 - 8*K1**2 - 8*K1*K2 - 8*K1 + 4*K2 + 5
2-strand cable arrow polynomial -128*K1**4*K2**2 + 256*K1**4*K2 - 736*K1**4 + 128*K1**3*K2*K3 - 256*K1**3*K3 - 768*K1**2*K2**4 + 2624*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 7008*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 6544*K1**2*K2 - 160*K1**2*K3**2 - 4264*K1**2 + 1728*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1536*K1*K2**2*K3 - 256*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 5504*K1*K2*K3 + 480*K1*K3*K4 + 16*K1*K4*K5 + 16*K1*K5*K6 - 256*K2**6 + 192*K2**4*K4 - 2880*K2**4 - 896*K2**2*K3**2 - 128*K2**2*K4**2 + 1872*K2**2*K4 - 1448*K2**2 + 224*K2*K3*K5 + 16*K2*K4*K6 - 1160*K3**2 - 304*K4**2 - 32*K5**2 - 8*K6**2 + 2942
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}]]
If K is slice True
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