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Flat knot 6.206

Min(phi) over symmetries of the knot is: [-3,-2,2,3,-1,2,4,1,2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.206', '6.214', '7.15858', '7.31209']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.206', '6.236', '6.575', '6.580', '6.613', '6.619', '6.810', '6.819', '6.831', '6.838', '6.957', '6.1018', '6.1028', '6.1046', '6.1073', '6.1279', '6.1507', '6.1532', '6.1556', '6.1639', '6.1688', '6.1924', '6.1931']
Outer characteristic polynomial of the knot is: t^5+65t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.206', '6.214']
2-strand cable arrow polynomial of the knot is: -64*K1**4 + 128*K1**2*K2**3 - 736*K1**2*K2**2 + 816*K1**2*K2 - 32*K1**2*K4**2 - 632*K1**2 + 64*K1*K2**3*K3 + 560*K1*K2*K3 + 80*K1*K3*K4 + 32*K1*K4*K5 - 64*K2**6 + 64*K2**4*K4 - 496*K2**4 - 64*K2**2*K3**2 - 48*K2**2*K4**2 + 320*K2**2*K4 - 168*K2**2 + 16*K2*K3*K5 + 16*K2*K4*K6 - 160*K3**2 - 92*K4**2 - 8*K5**2 + 466
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.206']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71626', 'vk6.71791', 'vk6.72211', 'vk6.72353', 'vk6.72617', 'vk6.72760', 'vk6.73074', 'vk6.73161', 'vk6.77246', 'vk6.77331', 'vk6.77579', 'vk6.77868', 'vk6.77912', 'vk6.78010', 'vk6.81121', 'vk6.81170', 'vk6.81242', 'vk6.81340', 'vk6.81530', 'vk6.82485', 'vk6.84382', 'vk6.85443', 'vk6.86925', 'vk6.87141', 'vk6.87302', 'vk6.87894', 'vk6.88097', 'vk6.88188', 'vk6.88672', 'vk6.88790']
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 O1O2O3O4O5U2O6U4U5U1U6U3
R3 orbit {'O1O2O3O4U1O5U3O6U2U5U4U6', 'O1O2O3O4U1O5O6U4U2U5U6U3', 'O1O2O3O4O5U2U3O6U5U1U4U6', 'O1O2O3O4O5U2O6U4U5U1U6U3'}
R3 orbit length 4
Gauss code of -K O1O2O3O4O5U3U6U5U1U2O6U4
Gauss code of K* O1O2O3O4O5U3U6U5U1U2O6U4
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 -2 -3 2 -1 1 3],[ 2 0 -2 3 0 2 3],[ 3 2 0 3 1 2 2],[-2 -3 -3 0 -2 0 2],[ 1 0 -1 2 0 1 2],[-1 -2 -2 0 -1 0 1],[-3 -3 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 3 2 -2 -3],[-3 0 -2 -3 -2],[-2 2 0 -3 -3],[ 2 3 3 0 -2],[ 3 2 3 2 0]]
If based matrix primitive False
Phi of primitive based matrix [-3,-2,2,3,2,3,2,3,3,2]
Phi over symmetry [-3,-2,2,3,-1,2,4,1,2,-1]
Phi of -K [-3,-2,2,3,-1,2,4,1,2,-1]
Phi of K* [-3,-2,2,3,-1,2,4,1,2,-1]
Phi of -K* [-3,-2,2,3,2,3,2,3,3,2]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z+9
Enhanced Jones-Krushkal polynomial -4w^3z+8w^2z+9w
Inner characteristic polynomial t^4+39t^2+1
Outer characteristic polynomial t^5+65t^3+5t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3
2-strand cable arrow polynomial -64*K1**4 + 128*K1**2*K2**3 - 736*K1**2*K2**2 + 816*K1**2*K2 - 32*K1**2*K4**2 - 632*K1**2 + 64*K1*K2**3*K3 + 560*K1*K2*K3 + 80*K1*K3*K4 + 32*K1*K4*K5 - 64*K2**6 + 64*K2**4*K4 - 496*K2**4 - 64*K2**2*K3**2 - 48*K2**2*K4**2 + 320*K2**2*K4 - 168*K2**2 + 16*K2*K3*K5 + 16*K2*K4*K6 - 160*K3**2 - 92*K4**2 - 8*K5**2 + 466
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}]]
If K is slice True
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