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Flat knot 6.546

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,-1,1,1,2,2,1,1,2,2,0,-1,-1,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.546']
Arrow polynomial of the knot is: -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.546', '6.591', '6.598', '6.666', '6.680', '6.742', '6.778', '6.805', '6.822', '6.824', '6.1129', '6.1512', '6.1647', '6.1678', '6.1705', '6.1847', '6.1857']
Outer characteristic polynomial of the knot is: t^7+36t^5+28t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.546']
2-strand cable arrow polynomial of the knot is: -768*K1**6 - 448*K1**4*K2**2 + 2144*K1**4*K2 - 6784*K1**4 + 672*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1280*K1**3*K3 - 6576*K1**2*K2**2 - 736*K1**2*K2*K4 + 12952*K1**2*K2 - 1664*K1**2*K3**2 - 432*K1**2*K4**2 - 5688*K1**2 - 832*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 9880*K1*K2*K3 + 2600*K1*K3*K4 + 272*K1*K4*K5 - 656*K2**4 - 656*K2**2*K3**2 - 176*K2**2*K4**2 + 1344*K2**2*K4 - 5660*K2**2 + 464*K2*K3*K5 + 96*K2*K4*K6 - 3032*K3**2 - 944*K4**2 - 88*K5**2 - 4*K6**2 + 5998
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.546']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4453', 'vk6.4550', 'vk6.5835', 'vk6.5964', 'vk6.7889', 'vk6.8005', 'vk6.9314', 'vk6.9435', 'vk6.13413', 'vk6.13508', 'vk6.13697', 'vk6.14079', 'vk6.15054', 'vk6.15176', 'vk6.17803', 'vk6.17834', 'vk6.18819', 'vk6.19439', 'vk6.19732', 'vk6.24346', 'vk6.25418', 'vk6.25449', 'vk6.26611', 'vk6.33255', 'vk6.33314', 'vk6.37546', 'vk6.44896', 'vk6.48642', 'vk6.50536', 'vk6.53647', 'vk6.55818', 'vk6.65482']
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 O1O2O3O4O5U3U5U1O6U4U6U2
R3 orbit {'O1O2O3O4O5U3U5U1O6U4U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U6U2O6U5U1U3
Gauss code of K* O1O2O3U4O5O4O6U3U6U1U5U2
Gauss code of -K* O1O2O3U2O4O5O6U5U3U6U1U4
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 1 -2 1 1 1],[ 2 0 2 -1 2 1 1],[-1 -2 0 -2 0 1 1],[ 2 1 2 0 2 1 1],[-1 -2 0 -2 0 0 1],[-1 -1 -1 -1 0 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 1 1 1 1 -2 -2],[-1 0 1 1 0 -2 -2],[-1 -1 0 0 0 -1 -1],[-1 -1 0 0 -1 -1 -1],[-1 0 0 1 0 -2 -2],[ 2 2 1 1 2 0 1],[ 2 2 1 1 2 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,2,2,-1,-1,0,2,2,0,0,1,1,1,1,1,2,2,-1]
Phi over symmetry [-2,-2,1,1,1,1,-1,1,1,2,2,1,1,2,2,0,-1,-1,-1,0,0]
Phi of -K [-2,-2,1,1,1,1,-1,1,1,2,2,1,1,2,2,0,-1,-1,-1,0,0]
Phi of K* [-1,-1,-1,-1,2,2,-1,-1,0,2,2,0,0,1,1,1,1,1,2,2,-1]
Phi of -K* [-2,-2,1,1,1,1,-1,1,1,2,2,1,1,2,2,0,-1,-1,-1,0,0]
Symmetry type of based matrix +
u-polynomial 2t^2-4t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+24t^4+14t^2+1
Outer characteristic polynomial t^7+36t^5+28t^3+5t
Flat arrow polynomial -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -768*K1**6 - 448*K1**4*K2**2 + 2144*K1**4*K2 - 6784*K1**4 + 672*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1280*K1**3*K3 - 6576*K1**2*K2**2 - 736*K1**2*K2*K4 + 12952*K1**2*K2 - 1664*K1**2*K3**2 - 432*K1**2*K4**2 - 5688*K1**2 - 832*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 9880*K1*K2*K3 + 2600*K1*K3*K4 + 272*K1*K4*K5 - 656*K2**4 - 656*K2**2*K3**2 - 176*K2**2*K4**2 + 1344*K2**2*K4 - 5660*K2**2 + 464*K2*K3*K5 + 96*K2*K4*K6 - 3032*K3**2 - 944*K4**2 - 88*K5**2 - 4*K6**2 + 5998
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {4}, {1, 2}]]
If K is slice False
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