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Flat knot 6.943

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,0,1,2,0,1,0,0,0,1,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.943', '7.16078']
Arrow polynomial of the knot is: 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.140', '6.569', '6.943', '6.970', '6.1234', '6.1298', '6.1311', '6.1326', '6.1500', '6.1506', '6.1708', '6.1712', '6.1720', '6.1859']
Outer characteristic polynomial of the knot is: t^7+34t^5+108t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.943', '7.16078']
2-strand cable arrow polynomial of the knot is: 2304*K1**4*K2 - 4672*K1**4 + 2048*K1**3*K2*K3 - 1152*K1**3*K3 - 384*K1**2*K2**4 + 1952*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 8400*K1**2*K2**2 - 864*K1**2*K2*K4 + 5320*K1**2*K2 - 2368*K1**2*K3**2 - 96*K1**2*K4**2 + 1072*K1**2 + 1728*K1*K2**3*K3 - 1248*K1*K2**2*K3 - 224*K1*K2**2*K5 - 576*K1*K2*K3*K4 + 5184*K1*K2*K3 + 1176*K1*K3*K4 + 56*K1*K4*K5 - 288*K2**6 + 448*K2**4*K4 - 2568*K2**4 - 32*K2**3*K6 - 1584*K2**2*K3**2 - 208*K2**2*K4**2 + 1488*K2**2*K4 + 530*K2**2 + 776*K2*K3*K5 + 48*K2*K4*K6 - 396*K3**2 - 126*K4**2 - 36*K5**2 - 2*K6**2 + 708
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.943']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.510', 'vk6.602', 'vk6.635', 'vk6.1010', 'vk6.1107', 'vk6.1148', 'vk6.1868', 'vk6.2291', 'vk6.2519', 'vk6.2567', 'vk6.2598', 'vk6.2807', 'vk6.2900', 'vk6.2923', 'vk6.3088', 'vk6.3208', 'vk6.5281', 'vk6.6541', 'vk6.8911', 'vk6.9827', 'vk6.20841', 'vk6.21117', 'vk6.22239', 'vk6.22549', 'vk6.28568', 'vk6.29802', 'vk6.39901', 'vk6.42222', 'vk6.46460', 'vk6.46843', 'vk6.48008', 'vk6.48082']
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 O1O2O3O4U5U6O5O6U4U1U3U2
R3 orbit {'O1O2O3O4U5U6O5O6U4U1U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U2U4U1O5O6U5U6
Gauss code of K* O1O2O3O4U2U4U3U1O5O6U5U6
Gauss code of -K* O1O2O3O4U5U6O5O6U4U2U1U3
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 1 0 -1 1],[ 2 0 2 1 0 1 3],[-1 -2 0 0 0 -2 0],[-1 -1 0 0 0 -2 0],[ 0 0 0 0 0 -1 1],[ 1 -1 2 2 1 0 1],[-1 -3 0 0 -1 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 0 -2 -1],[-1 0 0 0 0 -2 -2],[-1 0 0 0 -1 -1 -3],[ 0 0 0 1 0 -1 0],[ 1 2 2 1 1 0 -1],[ 2 1 2 3 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,0,0,0,2,1,0,0,2,2,1,1,3,1,0,1]
Phi over symmetry [-2,-1,0,1,1,1,0,2,0,1,2,0,1,0,0,0,1,1,0,0,0]
Phi of -K [-2,-1,0,1,1,1,0,2,0,1,2,0,1,0,0,0,1,1,0,0,0]
Phi of K* [-1,-1,-1,0,1,2,0,0,0,1,0,0,1,0,1,1,0,2,0,2,0]
Phi of -K* [-2,-1,0,1,1,1,1,0,1,2,3,1,2,2,1,0,0,1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial 6w^3z^2+19w^2z+15w
Inner characteristic polynomial t^6+26t^4+79t^2+4
Outer characteristic polynomial t^7+34t^5+108t^3+7t
Flat arrow polynomial 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
2-strand cable arrow polynomial 2304*K1**4*K2 - 4672*K1**4 + 2048*K1**3*K2*K3 - 1152*K1**3*K3 - 384*K1**2*K2**4 + 1952*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 8400*K1**2*K2**2 - 864*K1**2*K2*K4 + 5320*K1**2*K2 - 2368*K1**2*K3**2 - 96*K1**2*K4**2 + 1072*K1**2 + 1728*K1*K2**3*K3 - 1248*K1*K2**2*K3 - 224*K1*K2**2*K5 - 576*K1*K2*K3*K4 + 5184*K1*K2*K3 + 1176*K1*K3*K4 + 56*K1*K4*K5 - 288*K2**6 + 448*K2**4*K4 - 2568*K2**4 - 32*K2**3*K6 - 1584*K2**2*K3**2 - 208*K2**2*K4**2 + 1488*K2**2*K4 + 530*K2**2 + 776*K2*K3*K5 + 48*K2*K4*K6 - 396*K3**2 - 126*K4**2 - 36*K5**2 - 2*K6**2 + 708
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}]]
If K is slice False
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