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Flat knot 6.1703

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,0,1,1,1,1,0,0,1,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1703', '7.42929']
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+18t^5+29t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1703']
2-strand cable arrow polynomial of the knot is: -512*K1**6 - 768*K1**4*K2**2 + 3040*K1**4*K2 - 6720*K1**4 + 1952*K1**3*K2*K3 + 256*K1**3*K3*K4 - 1632*K1**3*K3 - 448*K1**2*K2**4 + 1888*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 10224*K1**2*K2**2 - 1536*K1**2*K2*K4 + 11280*K1**2*K2 - 1824*K1**2*K3**2 - 32*K1**2*K3*K5 - 400*K1**2*K4**2 - 2708*K1**2 + 1440*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 1440*K1*K2**2*K3 - 448*K1*K2**2*K5 - 736*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9224*K1*K2*K3 - 64*K1*K2*K4*K5 + 1936*K1*K3*K4 + 328*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1864*K2**4 - 32*K2**3*K6 - 1344*K2**2*K3**2 - 320*K2**2*K4**2 + 1848*K2**2*K4 - 3034*K2**2 - 96*K2*K3**2*K4 + 856*K2*K3*K5 + 208*K2*K4*K6 - 1868*K3**2 - 622*K4**2 - 104*K5**2 - 14*K6**2 + 3764
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1703']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16', 'vk6.27', 'vk6.29', 'vk6.144', 'vk6.150', 'vk6.161', 'vk6.167', 'vk6.1197', 'vk6.1203', 'vk6.1294', 'vk6.1305', 'vk6.1307', 'vk6.2361', 'vk6.2396', 'vk6.2402', 'vk6.2963', 'vk6.3532', 'vk6.3540', 'vk6.6916', 'vk6.6924', 'vk6.6949', 'vk6.6957', 'vk6.15376', 'vk6.15382', 'vk6.15497', 'vk6.33430', 'vk6.33447', 'vk6.33487', 'vk6.33502', 'vk6.33600', 'vk6.49929', 'vk6.53748']
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 O1O2O3U1U3O4O5U4U5O6U2U6
R3 orbit {'O1O2O3U1U3O4O5U4U5O6U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U2O4U5U6O5O6U1U3
Gauss code of K* O1O2U3O4O3U5U4U6O5O6U1U2
Gauss code of -K* O1O2U1O3O4U3U4O5O6U5U2U6
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 0 1 -1 1 1],[ 2 0 2 1 0 0 1],[ 0 -2 0 0 -1 1 1],[-1 -1 0 0 0 0 0],[ 1 0 1 0 0 1 0],[-1 0 -1 0 -1 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 0 0 -1],[-1 0 0 0 -1 0 -1],[-1 0 0 0 -1 -1 0],[ 0 0 1 1 0 -1 -2],[ 1 0 0 1 1 0 0],[ 2 1 1 0 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,0,0,0,0,1,0,1,0,1,1,1,0,1,2,0]
Phi over symmetry [-2,-1,0,1,1,1,0,2,0,1,1,1,1,0,0,1,0,1,0,0,0]
Phi of -K [-2,-1,0,1,1,1,1,0,2,2,3,0,2,2,1,0,1,0,0,0,0]
Phi of K* [-1,-1,-1,0,1,2,0,0,0,1,3,0,0,2,2,1,2,2,0,0,1]
Phi of -K* [-2,-1,0,1,1,1,0,2,0,1,1,1,1,0,0,1,0,1,0,0,0]
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+10t^4+12t^2+1
Outer characteristic polynomial t^7+18t^5+29t^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 -512*K1**6 - 768*K1**4*K2**2 + 3040*K1**4*K2 - 6720*K1**4 + 1952*K1**3*K2*K3 + 256*K1**3*K3*K4 - 1632*K1**3*K3 - 448*K1**2*K2**4 + 1888*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 10224*K1**2*K2**2 - 1536*K1**2*K2*K4 + 11280*K1**2*K2 - 1824*K1**2*K3**2 - 32*K1**2*K3*K5 - 400*K1**2*K4**2 - 2708*K1**2 + 1440*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 1440*K1*K2**2*K3 - 448*K1*K2**2*K5 - 736*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9224*K1*K2*K3 - 64*K1*K2*K4*K5 + 1936*K1*K3*K4 + 328*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1864*K2**4 - 32*K2**3*K6 - 1344*K2**2*K3**2 - 320*K2**2*K4**2 + 1848*K2**2*K4 - 3034*K2**2 - 96*K2*K3**2*K4 + 856*K2*K3*K5 + 208*K2*K4*K6 - 1868*K3**2 - 622*K4**2 - 104*K5**2 - 14*K6**2 + 3764
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {5}, {1, 4}, {2, 3}]]
If K is slice False
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