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Flat knot 6.687

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,0,1,3,4,0,0,0,1,0,1,1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.687']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 6*K1*K2 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.388', '6.687', '6.762']
Outer characteristic polynomial of the knot is: t^7+50t^5+41t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.687']
2-strand cable arrow polynomial of the knot is: -448*K1**6 - 448*K1**4*K2**2 + 3136*K1**4*K2 - 6288*K1**4 + 1024*K1**3*K2*K3 - 1408*K1**3*K3 - 192*K1**2*K2**4 + 1248*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 8960*K1**2*K2**2 - 1216*K1**2*K2*K4 + 13064*K1**2*K2 - 624*K1**2*K3**2 - 112*K1**2*K4**2 - 5988*K1**2 + 448*K1*K2**3*K3 - 1664*K1*K2**2*K3 - 256*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 9520*K1*K2*K3 - 32*K1*K2*K4*K5 + 1672*K1*K3*K4 + 264*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 1120*K2**4 - 384*K2**2*K3**2 - 56*K2**2*K4**2 + 1936*K2**2*K4 - 5756*K2**2 + 488*K2*K3*K5 + 56*K2*K4*K6 + 16*K3**2*K6 - 2644*K3**2 - 952*K4**2 - 200*K5**2 - 28*K6**2 + 5886
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.687']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4143', 'vk6.4176', 'vk6.5385', 'vk6.5418', 'vk6.7511', 'vk6.7536', 'vk6.9016', 'vk6.9049', 'vk6.12448', 'vk6.12481', 'vk6.13341', 'vk6.13560', 'vk6.13593', 'vk6.14249', 'vk6.14696', 'vk6.14724', 'vk6.15202', 'vk6.15856', 'vk6.15884', 'vk6.30849', 'vk6.30882', 'vk6.32037', 'vk6.32070', 'vk6.33057', 'vk6.33090', 'vk6.33853', 'vk6.34312', 'vk6.48487', 'vk6.50272', 'vk6.53529', 'vk6.53950', 'vk6.54261']
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 O1O2O3O4U3O5U2U5O6U1U6U4
R3 orbit {'O1O2O3O4U3O5U2U5O6U1U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U4O5U6U3O6U2
Gauss code of K* O1O2O3U1U4U5U3O5U6O4O6U2
Gauss code of -K* O1O2O3U2O4O5U4O6U1U6U5U3
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 -2 -1 3 1 1],[ 2 0 -1 0 4 1 1],[ 2 1 0 0 3 1 0],[ 1 0 0 0 1 0 0],[-3 -4 -3 -1 0 0 0],[-1 -1 -1 0 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 3 1 1 -1 -2 -2],[-3 0 0 0 -1 -3 -4],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 -1 -1],[ 1 1 0 0 0 0 0],[ 2 3 0 1 0 0 1],[ 2 4 1 1 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,2,2,0,0,1,3,4,0,0,0,1,0,1,1,0,0,-1]
Phi over symmetry [-3,-1,-1,1,2,2,0,0,1,3,4,0,0,0,1,0,1,1,0,0,-1]
Phi of -K [-2,-2,-1,1,1,3,-1,1,2,3,2,1,2,2,1,2,2,3,0,2,2]
Phi of K* [-3,-1,-1,1,2,2,2,2,3,1,2,0,2,2,2,2,2,3,1,1,-1]
Phi of -K* [-2,-2,-1,1,1,3,-1,0,1,1,4,0,0,1,3,0,0,1,0,0,0]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+30t^4+15t^2+1
Outer characteristic polynomial t^7+50t^5+41t^3+8t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 6*K1*K2 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -448*K1**6 - 448*K1**4*K2**2 + 3136*K1**4*K2 - 6288*K1**4 + 1024*K1**3*K2*K3 - 1408*K1**3*K3 - 192*K1**2*K2**4 + 1248*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 8960*K1**2*K2**2 - 1216*K1**2*K2*K4 + 13064*K1**2*K2 - 624*K1**2*K3**2 - 112*K1**2*K4**2 - 5988*K1**2 + 448*K1*K2**3*K3 - 1664*K1*K2**2*K3 - 256*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 9520*K1*K2*K3 - 32*K1*K2*K4*K5 + 1672*K1*K3*K4 + 264*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 1120*K2**4 - 384*K2**2*K3**2 - 56*K2**2*K4**2 + 1936*K2**2*K4 - 5756*K2**2 + 488*K2*K3*K5 + 56*K2*K4*K6 + 16*K3**2*K6 - 2644*K3**2 - 952*K4**2 - 200*K5**2 - 28*K6**2 + 5886
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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