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Flat knot 6.1739

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,2,1,1,2,0,0,1,1,1,0,1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1739', '7.38670']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.568', '6.806', '6.1000', '6.1049', '6.1081', '6.1101', '6.1112', '6.1122', '6.1193', '6.1195', '6.1208', '6.1235', '6.1263', '6.1517', '6.1528', '6.1537', '6.1542', '6.1545', '6.1558', '6.1569', '6.1575', '6.1644', '6.1650', '6.1681', '6.1692', '6.1702', '6.1706', '6.1728', '6.1734', '6.1739', '6.1799', '6.1813', '6.1820', '6.1834', '6.1840', '6.1851', '6.1861', '6.1878']
Outer characteristic polynomial of the knot is: t^7+24t^5+59t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1739', '7.38670']
2-strand cable arrow polynomial of the knot is: 3840*K1**4*K2 - 7328*K1**4 + 2048*K1**3*K2*K3 - 704*K1**3*K3 - 384*K1**2*K2**4 + 3008*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 12528*K1**2*K2**2 - 1024*K1**2*K2*K4 + 7416*K1**2*K2 - 1312*K1**2*K3**2 - 96*K1**2*K4**2 + 2128*K1**2 + 1728*K1*K2**3*K3 - 2464*K1*K2**2*K3 - 544*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 6176*K1*K2*K3 + 936*K1*K3*K4 + 56*K1*K4*K5 - 288*K2**6 + 448*K2**4*K4 - 3032*K2**4 - 128*K2**3*K6 - 1424*K2**2*K3**2 - 112*K2**2*K4**2 + 1848*K2**2*K4 + 522*K2**2 + 600*K2*K3*K5 + 48*K2*K4*K6 - 412*K3**2 - 146*K4**2 - 36*K5**2 - 2*K6**2 + 840
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1739']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.408', 'vk6.455', 'vk6.459', 'vk6.851', 'vk6.902', 'vk6.907', 'vk6.1610', 'vk6.2087', 'vk6.2472', 'vk6.2507', 'vk6.2510', 'vk6.2715', 'vk6.2760', 'vk6.2764', 'vk6.3026', 'vk6.3153', 'vk6.3304', 'vk6.3319', 'vk6.3485', 'vk6.3494', 'vk6.19903', 'vk6.19907', 'vk6.25828', 'vk6.25834', 'vk6.26347', 'vk6.26350', 'vk6.26790', 'vk6.26795', 'vk6.37940', 'vk6.37945', 'vk6.45087', 'vk6.45092']
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 O1O2O3U2U4O5O6U5U6O4U1U3
R3 orbit {'O1O2O3U2U4O5O6U5U6O4U1U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U3O4U5U6O5O6U4U2
Gauss code of K* O1O2U3O4O5U4U6U5O6O3U1U2
Gauss code of -K* O1O2U3O4O5U4U5O3O6U1U6U2
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 0 2 1 -1 1],[ 1 0 0 1 1 0 0],[-2 -2 -1 0 -2 -1 1],[ 0 -1 -1 2 0 0 0],[ 1 1 0 1 0 0 1],[-1 -1 0 -1 0 -1 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 1 -2 -1 -1 -2],[-1 -1 0 0 0 -1 -1],[ 0 2 0 0 -1 0 -1],[ 1 1 0 1 0 0 0],[ 1 1 1 0 0 0 1],[ 1 2 1 1 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,-1,2,1,1,2,0,0,1,1,1,0,1,0,0,-1]
Phi over symmetry [-2,-1,0,1,1,1,-1,2,1,1,2,0,0,1,1,1,0,1,0,0,-1]
Phi of -K [-1,-1,-1,0,1,2,-1,0,1,1,2,0,0,1,1,0,2,2,1,0,2]
Phi of K* [-2,-1,0,1,1,1,2,0,1,2,2,1,1,1,2,0,1,0,-1,0,0]
Phi of -K* [-1,-1,-1,0,1,2,-1,0,1,1,2,0,0,1,1,1,0,1,0,2,-1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial 6w^3z^2+19w^2z+15w
Inner characteristic polynomial t^6+16t^4+36t^2+1
Outer characteristic polynomial t^7+24t^5+59t^3+4t
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2
2-strand cable arrow polynomial 3840*K1**4*K2 - 7328*K1**4 + 2048*K1**3*K2*K3 - 704*K1**3*K3 - 384*K1**2*K2**4 + 3008*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 12528*K1**2*K2**2 - 1024*K1**2*K2*K4 + 7416*K1**2*K2 - 1312*K1**2*K3**2 - 96*K1**2*K4**2 + 2128*K1**2 + 1728*K1*K2**3*K3 - 2464*K1*K2**2*K3 - 544*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 6176*K1*K2*K3 + 936*K1*K3*K4 + 56*K1*K4*K5 - 288*K2**6 + 448*K2**4*K4 - 3032*K2**4 - 128*K2**3*K6 - 1424*K2**2*K3**2 - 112*K2**2*K4**2 + 1848*K2**2*K4 + 522*K2**2 + 600*K2*K3*K5 + 48*K2*K4*K6 - 412*K3**2 - 146*K4**2 - 36*K5**2 - 2*K6**2 + 840
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {5}, {2, 3}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {4, 5}, {2, 3}, {1}]]
If K is slice False
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