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Flat knot 6.15

Min(phi) over symmetries of the knot is: [-5,-2,-2,3,3,3,1,2,3,4,5,0,1,2,3,2,3,4,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.15']
Arrow polynomial of the knot is: -2*K1*K4 + K3 + K5 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.4', '6.12', '6.15', '6.48']
Outer characteristic polynomial of the knot is: t^7+158t^5+261t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.15']
2-strand cable arrow polynomial of the knot is: -32*K1**2 + 336*K1*K2*K3 - 2*K10**2 + 8*K10*K2*K8 - 32*K2**4 + 64*K2**3*K3*K5 - 576*K2**2*K3**2 - 128*K2**2*K5**2 - 8*K2**2*K8**2 - 532*K2**2 + 1344*K2*K3*K5 + 48*K2*K5*K7 + 40*K2*K6*K8 - 568*K3**2 + 48*K3*K5*K8 - 536*K5**2 - 18*K6**2 - 44*K8**2 + 586
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.15']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81850', 'vk6.81897', 'vk6.82076', 'vk6.82091', 'vk6.82782', 'vk6.82789', 'vk6.82851', 'vk6.82953', 'vk6.83292', 'vk6.83392', 'vk6.83456', 'vk6.84554', 'vk6.84648', 'vk6.84781', 'vk6.84786', 'vk6.86276', 'vk6.86847', 'vk6.88470', 'vk6.88543', 'vk6.90032']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3O4O5O6U1U3U2U6U5U4
R3 orbit {'O1O2O3O4O5O6U1U3U2U6U5U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U3U2U1U5U4U6
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U3U2U1U5U4U6
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 -5 -2 -2 3 3 3],[ 5 0 2 1 5 4 3],[ 2 -2 0 0 4 3 2],[ 2 -1 0 0 3 2 1],[-3 -5 -4 -3 0 0 0],[-3 -4 -3 -2 0 0 0],[-3 -3 -2 -1 0 0 0]]
Primitive based matrix [[ 0 3 3 3 -2 -2 -5],[-3 0 0 0 -1 -2 -3],[-3 0 0 0 -2 -3 -4],[-3 0 0 0 -3 -4 -5],[ 2 1 2 3 0 0 -1],[ 2 2 3 4 0 0 -2],[ 5 3 4 5 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-3,2,2,5,0,0,1,2,3,0,2,3,4,3,4,5,0,1,2]
Phi over symmetry [-5,-2,-2,3,3,3,1,2,3,4,5,0,1,2,3,2,3,4,0,0,0]
Phi of -K [-5,-2,-2,3,3,3,1,2,3,4,5,0,1,2,3,2,3,4,0,0,0]
Phi of K* [-3,-3,-3,2,2,5,0,0,1,2,3,0,2,3,4,3,4,5,0,1,2]
Phi of -K* [-5,-2,-2,3,3,3,1,2,3,4,5,0,1,2,3,2,3,4,0,0,0]
Symmetry type of based matrix +
u-polynomial t^5-3t^3+2t^2
Normalized Jones-Krushkal polynomial z+3
Enhanced Jones-Krushkal polynomial -8w^5z+8w^4z-8w^3z+9w^2z+3w
Inner characteristic polynomial t^6+98t^4+41t^2
Outer characteristic polynomial t^7+158t^5+261t^3
Flat arrow polynomial -2*K1*K4 + K3 + K5 + 1
2-strand cable arrow polynomial -32*K1**2 + 336*K1*K2*K3 - 2*K10**2 + 8*K10*K2*K8 - 32*K2**4 + 64*K2**3*K3*K5 - 576*K2**2*K3**2 - 128*K2**2*K5**2 - 8*K2**2*K8**2 - 532*K2**2 + 1344*K2*K3*K5 + 48*K2*K5*K7 + 40*K2*K6*K8 - 568*K3**2 + 48*K3*K5*K8 - 536*K5**2 - 18*K6**2 - 44*K8**2 + 586
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {5}, {2, 3}, {1}]]
If K is slice False
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