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Flat knot 6.1010

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,1,1,2,1,3,0,1,2,2,0,1,1,0,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1010']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 6*K1*K2 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.552', '6.652', '6.764', '6.776', '6.784', '6.839', '6.903', '6.1010', '6.1166']
Outer characteristic polynomial of the knot is: t^7+55t^5+76t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1010']
2-strand cable arrow polynomial of the knot is: -16*K1**4 + 32*K1**3*K2*K3 - 64*K1**3*K3 - 256*K1**2*K2**4 + 608*K1**2*K2**3 - 4112*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 128*K1**2*K2*K4 + 4656*K1**2*K2 - 96*K1**2*K3**2 - 16*K1**2*K4**2 - 4148*K1**2 + 1024*K1*K2**3*K3 - 1120*K1*K2**2*K3 - 128*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 5496*K1*K2*K3 + 560*K1*K3*K4 + 168*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1360*K2**4 - 32*K2**3*K6 - 608*K2**2*K3**2 - 24*K2**2*K4**2 + 1384*K2**2*K4 - 2704*K2**2 - 32*K2*K3**2*K4 + 352*K2*K3*K5 + 32*K2*K4*K6 - 16*K3**4 + 24*K3**2*K6 - 1784*K3**2 - 436*K4**2 - 108*K5**2 - 8*K6**2 + 3114
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1010']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71467', 'vk6.71519', 'vk6.71529', 'vk6.71996', 'vk6.72008', 'vk6.72049', 'vk6.72059', 'vk6.73218', 'vk6.73227', 'vk6.73249', 'vk6.73260', 'vk6.73661', 'vk6.73672', 'vk6.75141', 'vk6.75158', 'vk6.77094', 'vk6.77143', 'vk6.77147', 'vk6.77440', 'vk6.77442', 'vk6.78076', 'vk6.78081', 'vk6.78108', 'vk6.78117', 'vk6.81292', 'vk6.81540', 'vk6.81554', 'vk6.85474', 'vk6.85476', 'vk6.86890', 'vk6.87730', 'vk6.89510']
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 O1O2O3O4U2U5O6U3O5U1U6U4
R3 orbit {'O1O2O3O4U2U5O6U3O5U1U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U4O6U2O5U6U3
Gauss code of K* O1O2O3U1U4U5U3O4O6U2O5U6
Gauss code of -K* O1O2O3U4O5U2O4O6U1U5U6U3
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 0 3 0 1],[ 2 0 -1 2 4 1 1],[ 2 1 0 1 2 1 1],[ 0 -2 -1 0 1 0 0],[-3 -4 -2 -1 0 -2 -1],[ 0 -1 -1 0 2 0 1],[-1 -1 -1 0 1 -1 0]]
Primitive based matrix [[ 0 3 1 0 0 -2 -2],[-3 0 -1 -1 -2 -2 -4],[-1 1 0 0 -1 -1 -1],[ 0 1 0 0 0 -1 -2],[ 0 2 1 0 0 -1 -1],[ 2 2 1 1 1 0 1],[ 2 4 1 2 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,2,2,1,1,2,2,4,0,1,1,1,0,1,2,1,1,-1]
Phi over symmetry [-3,-1,0,0,2,2,1,1,2,1,3,0,1,2,2,0,1,1,0,1,-1]
Phi of -K [-2,-2,0,0,1,3,-1,1,1,2,3,0,1,2,1,0,1,2,0,1,1]
Phi of K* [-3,-1,0,0,2,2,1,1,2,1,3,0,1,2,2,0,1,1,0,1,-1]
Phi of -K* [-2,-2,0,0,1,3,-1,1,2,1,4,1,1,1,2,0,1,2,0,1,1]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial 6w^3z^2+23w^2z+23w
Inner characteristic polynomial t^6+37t^4+32t^2
Outer characteristic polynomial t^7+55t^5+76t^3+4t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 6*K1*K2 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial -16*K1**4 + 32*K1**3*K2*K3 - 64*K1**3*K3 - 256*K1**2*K2**4 + 608*K1**2*K2**3 - 4112*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 128*K1**2*K2*K4 + 4656*K1**2*K2 - 96*K1**2*K3**2 - 16*K1**2*K4**2 - 4148*K1**2 + 1024*K1*K2**3*K3 - 1120*K1*K2**2*K3 - 128*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 5496*K1*K2*K3 + 560*K1*K3*K4 + 168*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1360*K2**4 - 32*K2**3*K6 - 608*K2**2*K3**2 - 24*K2**2*K4**2 + 1384*K2**2*K4 - 2704*K2**2 - 32*K2*K3**2*K4 + 352*K2*K3*K5 + 32*K2*K4*K6 - 16*K3**4 + 24*K3**2*K6 - 1784*K3**2 - 436*K4**2 - 108*K5**2 - 8*K6**2 + 3114
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}]]
If K is slice False
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