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Flat knot 6.374

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,0,1,2,2,1,1,1,1,1,0,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.374', '7.25963']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.374', '6.446', '6.527', '6.1218', '6.1237', '6.1276', '6.1498', '6.1523', '6.1595', '6.1703', '6.1751', '6.1766', '6.1849', '6.1926']
Outer characteristic polynomial of the knot is: t^7+28t^5+43t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.374', '6.1511', '6.1662']
2-strand cable arrow polynomial of the knot is: -1024*K1**6 - 1728*K1**4*K2**2 + 3488*K1**4*K2 - 5776*K1**4 + 2272*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1440*K1**3*K3 - 832*K1**2*K2**4 + 3104*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 11136*K1**2*K2**2 - 1408*K1**2*K2*K4 + 11024*K1**2*K2 - 1648*K1**2*K3**2 - 384*K1**2*K4**2 - 2780*K1**2 + 2208*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 2016*K1*K2**2*K3 - 512*K1*K2**2*K5 - 576*K1*K2*K3*K4 + 9384*K1*K2*K3 - 32*K1*K2*K4*K5 + 1864*K1*K3*K4 + 368*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 2776*K2**4 - 32*K2**3*K6 - 1872*K2**2*K3**2 - 384*K2**2*K4**2 + 2304*K2**2*K4 - 2418*K2**2 - 128*K2*K3**2*K4 + 1016*K2*K3*K5 + 264*K2*K4*K6 - 1756*K3**2 - 618*K4**2 - 128*K5**2 - 22*K6**2 + 3616
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.374']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71', 'vk6.126', 'vk6.221', 'vk6.268', 'vk6.302', 'vk6.684', 'vk6.1223', 'vk6.1270', 'vk6.1359', 'vk6.1406', 'vk6.1446', 'vk6.1928', 'vk6.2379', 'vk6.2439', 'vk6.2929', 'vk6.2981', 'vk6.5739', 'vk6.5772', 'vk6.7804', 'vk6.7837', 'vk6.13272', 'vk6.13303', 'vk6.14786', 'vk6.14810', 'vk6.15942', 'vk6.15966', 'vk6.18058', 'vk6.24498', 'vk6.33029', 'vk6.33387', 'vk6.43918', 'vk6.50489']
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 O1O2O3O4O5U3U5O6U4U6U1U2
R3 orbit {'O1O2O3O4O5U3U5O6U4U6U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U5U6U2O6U1U3
Gauss code of K* O1O2O3O4U3U4U5U1U6O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U5U4U6U1U2
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 -1 1 -2 0 1 1],[ 1 0 1 -2 0 1 1],[-1 -1 0 -2 0 1 1],[ 2 2 2 0 2 1 1],[ 0 0 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 0 -1 -2],[-1 0 1 1 0 -1 -2],[-1 -1 0 0 0 -1 -1],[-1 -1 0 0 -1 -1 -1],[ 0 0 0 1 0 0 -2],[ 1 1 1 1 0 0 -2],[ 2 2 1 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,-1,0,1,2,0,0,1,1,1,1,1,0,2,2]
Phi over symmetry [-2,-1,0,1,1,1,-1,0,1,2,2,1,1,1,1,1,0,1,-1,-1,0]
Phi of -K [-2,-1,0,1,1,1,-1,0,1,2,2,1,1,1,1,1,0,1,-1,-1,0]
Phi of K* [-1,-1,-1,0,1,2,-1,0,0,1,2,1,1,1,1,1,1,2,1,0,-1]
Phi of -K* [-2,-1,0,1,1,1,2,2,1,1,2,0,1,1,1,0,1,0,0,-1,-1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+20t^4+18t^2+1
Outer characteristic polynomial t^7+28t^5+43t^3+8t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
2-strand cable arrow polynomial -1024*K1**6 - 1728*K1**4*K2**2 + 3488*K1**4*K2 - 5776*K1**4 + 2272*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1440*K1**3*K3 - 832*K1**2*K2**4 + 3104*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 11136*K1**2*K2**2 - 1408*K1**2*K2*K4 + 11024*K1**2*K2 - 1648*K1**2*K3**2 - 384*K1**2*K4**2 - 2780*K1**2 + 2208*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 2016*K1*K2**2*K3 - 512*K1*K2**2*K5 - 576*K1*K2*K3*K4 + 9384*K1*K2*K3 - 32*K1*K2*K4*K5 + 1864*K1*K3*K4 + 368*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 2776*K2**4 - 32*K2**3*K6 - 1872*K2**2*K3**2 - 384*K2**2*K4**2 + 2304*K2**2*K4 - 2418*K2**2 - 128*K2*K3**2*K4 + 1016*K2*K3*K5 + 264*K2*K4*K6 - 1756*K3**2 - 618*K4**2 - 128*K5**2 - 22*K6**2 + 3616
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice False
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