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Flat knot 6.1368

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,1,1,2,2,3,1,1,1,1,-1,-1,0,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1368']
Arrow polynomial of the knot is: 8*K1**3 - 6*K1*K2 - 3*K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.67', '6.535', '6.1347', '6.1348', '6.1368']
Outer characteristic polynomial of the knot is: t^7+41t^5+78t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1368']
2-strand cable arrow polynomial of the knot is: -3200*K1**2*K2**4 + 1792*K1**2*K2**3 - 2720*K1**2*K2**2 + 1312*K1**2*K2 - 696*K1**2 + 2368*K1*K2**3*K3 + 1936*K1*K2*K3 + 96*K1*K3*K4 - 832*K2**6 + 640*K2**4*K4 - 864*K2**4 - 480*K2**2*K3**2 - 152*K2**2*K4**2 + 536*K2**2*K4 + 122*K2**2 + 48*K2*K3*K5 + 24*K2*K4*K6 - 456*K3**2 - 164*K4**2 - 2*K6**2 + 690
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1368']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20133', 'vk6.20137', 'vk6.21425', 'vk6.21429', 'vk6.27237', 'vk6.27245', 'vk6.28901', 'vk6.28908', 'vk6.38649', 'vk6.38663', 'vk6.40862', 'vk6.40874', 'vk6.47252', 'vk6.47259', 'vk6.56958', 'vk6.56966', 'vk6.58114', 'vk6.58120', 'vk6.62656', 'vk6.62665', 'vk6.67453', 'vk6.67459', 'vk6.70027', 'vk6.70030']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3U4O5O4U1U2U3O6U5U6
R3 orbit {'O1O2O3U4O5O4U1U2U3O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4U1U2U3O6O5U6
Gauss code of K* Same
Gauss code of -K* O1O2O3U4U5O4U1U2U3O6O5U6
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 -3 -1 1 1 1 1],[ 3 0 1 2 3 2 1],[ 1 -1 0 1 1 1 1],[-1 -2 -1 0 -1 0 1],[-1 -3 -1 1 0 1 0],[-1 -2 -1 0 -1 0 1],[-1 -1 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 1 1 0 -1 -3],[-1 -1 0 0 1 -1 -2],[-1 -1 0 0 1 -1 -2],[-1 0 -1 -1 0 -1 -1],[ 1 1 1 1 1 0 -1],[ 3 3 2 2 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,1,3,-1,-1,0,1,3,0,-1,1,2,-1,1,2,1,1,1]
Phi over symmetry [-3,-1,1,1,1,1,1,1,2,2,3,1,1,1,1,-1,-1,0,0,-1,-1]
Phi of -K [-3,-1,1,1,1,1,1,1,2,2,3,1,1,1,1,-1,-1,0,0,-1,-1]
Phi of K* [-1,-1,-1,-1,1,3,-1,-1,0,1,3,0,-1,1,2,-1,1,2,1,1,1]
Phi of -K* [-3,-1,1,1,1,1,1,1,2,2,3,1,1,1,1,-1,-1,0,0,-1,-1]
Symmetry type of based matrix +
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial z+3
Enhanced Jones-Krushkal polynomial -12w^3z+13w^2z+3w
Inner characteristic polynomial t^6+27t^4+44t^2
Outer characteristic polynomial t^7+41t^5+78t^3
Flat arrow polynomial 8*K1**3 - 6*K1*K2 - 3*K1 + K3 + 1
2-strand cable arrow polynomial -3200*K1**2*K2**4 + 1792*K1**2*K2**3 - 2720*K1**2*K2**2 + 1312*K1**2*K2 - 696*K1**2 + 2368*K1*K2**3*K3 + 1936*K1*K2*K3 + 96*K1*K3*K4 - 832*K2**6 + 640*K2**4*K4 - 864*K2**4 - 480*K2**2*K3**2 - 152*K2**2*K4**2 + 536*K2**2*K4 + 122*K2**2 + 48*K2*K3*K5 + 24*K2*K4*K6 - 456*K3**2 - 164*K4**2 - 2*K6**2 + 690
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{4, 6}, {5}, {3}, {2}, {1}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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