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Flat knot 6.348

Min(phi) over symmetries of the knot is: [-4,-2,0,2,2,2,0,2,2,3,5,1,1,1,2,1,2,2,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.348']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K2**2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.92', '6.348', '6.490', '6.494']
Outer characteristic polynomial of the knot is: t^7+92t^5+80t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.348']
2-strand cable arrow polynomial of the knot is: -32*K2**4*K4**2 + 160*K2**4*K4 - 1728*K2**4 + 32*K2**2*K4**3 - 368*K2**2*K4**2 + 2248*K2**2*K4 - 532*K2**2 + 296*K2*K4*K6 - 8*K4**4 - 728*K4**2 - 76*K6**2 + 734
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.348']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72584', 'vk6.72585', 'vk6.72701', 'vk6.72702', 'vk6.73013', 'vk6.73014', 'vk6.73159', 'vk6.73160', 'vk6.73576', 'vk6.73590', 'vk6.74291', 'vk6.74293', 'vk6.74915', 'vk6.74917', 'vk6.75338', 'vk6.75357', 'vk6.76471', 'vk6.76473', 'vk6.77866', 'vk6.77910', 'vk6.77987', 'vk6.78009', 'vk6.79332', 'vk6.79334', 'vk6.80103', 'vk6.80110', 'vk6.80793', 'vk6.80795', 'vk6.85075', 'vk6.86707', 'vk6.87366', 'vk6.90160']
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 O1O2O3O4O5U3U2O6U4U1U6U5
R3 orbit {'O1O2O3O4O5U3U2O6U4U1U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U6U5U2O6U4U3
Gauss code of K* O1O2O3O4U2U5U6U1U4O6O5U3
Gauss code of -K* O1O2O3O4U2O5O6U1U4U6U5U3
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable True
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -2 -2 0 4 2],[ 2 0 -1 -1 2 5 2],[ 2 1 0 0 2 3 1],[ 2 1 0 0 1 2 1],[ 0 -2 -2 -1 0 2 1],[-4 -5 -3 -2 -2 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 4 2 0 -2 -2 -2],[-4 0 0 -2 -2 -3 -5],[-2 0 0 -1 -1 -1 -2],[ 0 2 1 0 -1 -2 -2],[ 2 2 1 1 0 0 1],[ 2 3 1 2 0 0 1],[ 2 5 2 2 -1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,0,2,2,2,0,2,2,3,5,1,1,1,2,1,2,2,0,-1,-1]
Phi over symmetry [-4,-2,0,2,2,2,0,2,2,3,5,1,1,1,2,1,2,2,0,-1,-1]
Phi of -K [-2,-2,-2,0,2,4,-1,0,0,3,3,1,0,2,1,1,3,4,1,2,2]
Phi of K* [-4,-2,0,2,2,2,2,2,1,3,4,1,2,3,3,0,0,1,-1,-1,0]
Phi of -K* [-2,-2,-2,0,2,4,-1,-1,2,2,5,0,1,1,2,2,1,3,1,2,0]
Symmetry type of based matrix c
u-polynomial -t^4+2t^2
Normalized Jones-Krushkal polynomial 7z^2+20z+13
Enhanced Jones-Krushkal polynomial -2w^4z^2+9w^3z^2+20w^2z+13
Inner characteristic polynomial t^6+60t^4+16t^2
Outer characteristic polynomial t^7+92t^5+80t^3
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K2**2 + 3
2-strand cable arrow polynomial -32*K2**4*K4**2 + 160*K2**4*K4 - 1728*K2**4 + 32*K2**2*K4**3 - 368*K2**2*K4**2 + 2248*K2**2*K4 - 532*K2**2 + 296*K2*K4*K6 - 8*K4**4 - 728*K4**2 - 76*K6**2 + 734
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}], [{3, 6}, {4, 5}, {2}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {2, 5}, {1, 4}, {3}]]
If K is slice False
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