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Flat knot 6.1143

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,1,1,1,2,2,0,1,0,1,1,-1,0,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1143']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.115', '6.407', '6.413', '6.448', '6.844', '6.879', '6.888', '6.926', '6.934', '6.1140', '6.1143', '6.1161', '6.1177']
Outer characteristic polynomial of the knot is: t^7+55t^5+83t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1143']
2-strand cable arrow polynomial of the knot is: 64*K1**4*K2 - 480*K1**4 + 384*K1**3*K2*K3 - 448*K1**3*K3 + 1472*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 4656*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 512*K1**2*K2*K4 + 5360*K1**2*K2 - 800*K1**2*K3**2 - 4564*K1**2 - 256*K1*K2**4*K3 + 1120*K1*K2**3*K3 + 640*K1*K2**2*K3*K4 - 1792*K1*K2**2*K3 - 224*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 256*K1*K2*K3*K6 + 6624*K1*K2*K3 + 1432*K1*K3*K4 + 40*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 384*K2**4*K4 - 2464*K2**4 + 32*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 1536*K2**2*K3**2 - 552*K2**2*K4**2 + 2328*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 2758*K2**2 + 1008*K2*K3*K5 + 312*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 2144*K3**2 - 784*K4**2 - 152*K5**2 - 58*K6**2 - 4*K7**2 - 2*K8**2 + 3688
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1143']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11507', 'vk6.11828', 'vk6.12843', 'vk6.13162', 'vk6.20274', 'vk6.21599', 'vk6.27544', 'vk6.29112', 'vk6.31272', 'vk6.31644', 'vk6.32416', 'vk6.32843', 'vk6.38945', 'vk6.41180', 'vk6.45716', 'vk6.47421', 'vk6.52274', 'vk6.52531', 'vk6.53105', 'vk6.53429', 'vk6.57107', 'vk6.58285', 'vk6.61692', 'vk6.62845', 'vk6.63793', 'vk6.63917', 'vk6.64231', 'vk6.64435', 'vk6.66738', 'vk6.67614', 'vk6.69392', 'vk6.70122']
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 O1O2O3O4U1U2U5O6O5U4U3U6
R3 orbit {'O1O2O3O4U1U2U5O6O5U4U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2U1O6O5U6U3U4
Gauss code of K* O1O2O3U4U3O5O6O4U1U2U6U5
Gauss code of -K* O1O2O3U4U1O4O5O6U3U2U5U6
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 -3 -1 1 1 1 1],[ 3 0 1 3 2 3 2],[ 1 -1 0 2 1 1 2],[-1 -3 -2 0 0 -1 1],[-1 -2 -1 0 0 -1 0],[-1 -3 -1 1 1 0 1],[-1 -2 -2 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 1 1 1 -1 -3],[-1 -1 0 1 0 -2 -3],[-1 -1 -1 0 0 -2 -2],[-1 -1 0 0 0 -1 -2],[ 1 1 2 2 1 0 -1],[ 3 3 3 2 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,1,3,-1,-1,-1,1,3,-1,0,2,3,0,2,2,1,2,1]
Phi over symmetry [-3,-1,1,1,1,1,1,1,1,2,2,0,1,0,1,1,-1,0,-1,-1,0]
Phi of -K [-3,-1,1,1,1,1,1,1,1,2,2,0,1,0,1,1,-1,0,-1,-1,0]
Phi of K* [-1,-1,-1,-1,1,3,-1,-1,0,0,2,-1,0,0,1,1,1,1,1,2,1]
Phi of -K* [-3,-1,1,1,1,1,1,2,2,3,3,1,2,1,2,0,-1,0,-1,-1,1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2-4w^3z+27w^2z+23w
Inner characteristic polynomial t^6+41t^4+33t^2+1
Outer characteristic polynomial t^7+55t^5+83t^3+13t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial 64*K1**4*K2 - 480*K1**4 + 384*K1**3*K2*K3 - 448*K1**3*K3 + 1472*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 4656*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 512*K1**2*K2*K4 + 5360*K1**2*K2 - 800*K1**2*K3**2 - 4564*K1**2 - 256*K1*K2**4*K3 + 1120*K1*K2**3*K3 + 640*K1*K2**2*K3*K4 - 1792*K1*K2**2*K3 - 224*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 256*K1*K2*K3*K6 + 6624*K1*K2*K3 + 1432*K1*K3*K4 + 40*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 384*K2**4*K4 - 2464*K2**4 + 32*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 1536*K2**2*K3**2 - 552*K2**2*K4**2 + 2328*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 2758*K2**2 + 1008*K2*K3*K5 + 312*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 2144*K3**2 - 784*K4**2 - 152*K5**2 - 58*K6**2 - 4*K7**2 - 2*K8**2 + 3688
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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