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Flat knot 6.117

Min(phi) over symmetries of the knot is: [-3,-1,0,0,1,3,-1,1,2,1,3,1,1,1,1,0,1,1,1,2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.117', '7.10417']
Arrow polynomial of the knot is: -4*K1*K3 + 2*K2 + 2*K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.117', '6.935']
Outer characteristic polynomial of the knot is: t^7+76t^5+75t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.117']
2-strand cable arrow polynomial of the knot is: -2944*K1**2*K3**2 - 224*K1**2*K6**2 - 632*K1**2 + 3488*K1*K2*K3 + 1760*K1*K3*K4 + 160*K1*K5*K6 + 128*K1*K6*K7 - 2208*K2**2*K3**2 - 176*K2**2*K6**2 - 544*K2**2 + 1248*K2*K3*K5 + 112*K2*K4*K6 + 96*K2*K6*K8 - 672*K3**2 - 156*K4**2 - 96*K5**2 - 48*K6**2 - 8*K7**2 - 4*K8**2 + 718
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.117']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.476', 'vk6.540', 'vk6.940', 'vk6.1038', 'vk6.1628', 'vk6.1729', 'vk6.2120', 'vk6.2222', 'vk6.2533', 'vk6.2831', 'vk6.3031', 'vk6.3164', 'vk6.20391', 'vk6.21732', 'vk6.27719', 'vk6.29263', 'vk6.39163', 'vk6.45887', 'vk6.57253', 'vk6.61888']
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 O1O2O3O4O5O6U3U5U4U6U1U2
R3 orbit {'O1O2O3O4O5O6U3U5U4U6U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U6U1U3U2U4
Gauss code of K* O1O2O3O4O5O6U5U6U1U3U2U4
Gauss code of -K* Same
Diagrammatic symmetry type -
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 1 -3 0 0 3],[ 1 0 1 -3 0 0 3],[-1 -1 0 -3 0 0 3],[ 3 3 3 0 2 1 3],[ 0 0 0 -2 0 0 2],[ 0 0 0 -1 0 0 1],[-3 -3 -3 -3 -2 -1 0]]
Primitive based matrix [[ 0 3 1 0 0 -1 -3],[-3 0 -3 -1 -2 -3 -3],[-1 3 0 0 0 -1 -3],[ 0 1 0 0 0 0 -1],[ 0 2 0 0 0 0 -2],[ 1 3 1 0 0 0 -3],[ 3 3 3 1 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,1,3,3,1,2,3,3,0,0,1,3,0,0,1,0,2,3]
Phi over symmetry [-3,-1,0,0,1,3,-1,1,2,1,3,1,1,1,1,0,1,1,1,2,-1]
Phi of -K [-3,-1,0,0,1,3,-1,1,2,1,3,1,1,1,1,0,1,1,1,2,-1]
Phi of K* [-3,-1,0,0,1,3,-1,1,2,1,3,1,1,1,1,0,1,1,1,2,-1]
Phi of -K* [-3,-1,0,0,1,3,3,1,2,3,3,0,0,1,3,0,0,1,0,2,3]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z+15
Enhanced Jones-Krushkal polynomial 4w^4z-6w^3z+9w^2z+15w
Inner characteristic polynomial t^6+56t^4+19t^2
Outer characteristic polynomial t^7+76t^5+75t^3
Flat arrow polynomial -4*K1*K3 + 2*K2 + 2*K4 + 1
2-strand cable arrow polynomial -2944*K1**2*K3**2 - 224*K1**2*K6**2 - 632*K1**2 + 3488*K1*K2*K3 + 1760*K1*K3*K4 + 160*K1*K5*K6 + 128*K1*K6*K7 - 2208*K2**2*K3**2 - 176*K2**2*K6**2 - 544*K2**2 + 1248*K2*K3*K5 + 112*K2*K4*K6 + 96*K2*K6*K8 - 672*K3**2 - 156*K4**2 - 96*K5**2 - 48*K6**2 - 8*K7**2 - 4*K8**2 + 718
Genus of based matrix 0
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {4}, {1, 2}]]
If K is slice True
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