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Flat knot 6.1743

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,1,1,0,1,1,1,1,0,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1743']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.906', '6.1223', '6.1338', '6.1351', '6.1571', '6.1670', '6.1718', '6.1743', '6.1765', '6.1793', '6.1852', '6.2070']
Outer characteristic polynomial of the knot is: t^7+14t^5+20t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1743']
2-strand cable arrow polynomial of the knot is: -704*K1**6 - 576*K1**4*K2**2 + 3808*K1**4*K2 - 8288*K1**4 + 1568*K1**3*K2*K3 - 1248*K1**3*K3 - 192*K1**2*K2**4 + 1280*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 10576*K1**2*K2**2 - 1184*K1**2*K2*K4 + 14016*K1**2*K2 - 640*K1**2*K3**2 - 112*K1**2*K4**2 - 3804*K1**2 + 608*K1*K2**3*K3 - 1376*K1*K2**2*K3 - 448*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 8336*K1*K2*K3 + 824*K1*K3*K4 + 104*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 1344*K2**4 - 96*K2**3*K6 - 336*K2**2*K3**2 - 128*K2**2*K4**2 + 1416*K2**2*K4 - 4418*K2**2 + 272*K2*K3*K5 + 72*K2*K4*K6 - 1576*K3**2 - 312*K4**2 - 44*K5**2 - 6*K6**2 + 4622
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1743']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4137', 'vk6.4170', 'vk6.5375', 'vk6.5408', 'vk6.7505', 'vk6.7532', 'vk6.9006', 'vk6.9039', 'vk6.12435', 'vk6.12466', 'vk6.13345', 'vk6.13564', 'vk6.13597', 'vk6.14254', 'vk6.14703', 'vk6.14727', 'vk6.15203', 'vk6.15857', 'vk6.15881', 'vk6.30848', 'vk6.30879', 'vk6.32032', 'vk6.32063', 'vk6.33069', 'vk6.33102', 'vk6.33850', 'vk6.34313', 'vk6.48495', 'vk6.50280', 'vk6.53517', 'vk6.53947', 'vk6.54262']
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 O1O2O3U4U1O4O5U3U5O6U2U6
R3 orbit {'O1O2O3U4U1O4O5U3U5O6U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U2O4U5U1O5O6U3U6
Gauss code of K* O1O2U3O4O3U5U4U1O6O5U6U2
Gauss code of -K* O1O2U1O3O4U3U5O6O5U4U2U6
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 0 0 -1 1 1],[ 1 0 1 0 1 1 1],[ 0 -1 0 -1 0 1 1],[ 0 0 1 0 0 1 0],[ 1 -1 0 0 0 1 1],[-1 -1 -1 -1 -1 0 0],[-1 -1 -1 0 -1 0 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 0 0 -1 -1 -1],[-1 0 0 -1 -1 -1 -1],[ 0 0 1 0 1 0 0],[ 0 1 1 -1 0 0 -1],[ 1 1 1 0 0 0 -1],[ 1 1 1 0 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,0,0,1,1,1,1,1,1,1,-1,0,0,0,1,1]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,0,1,1,0,1,1,1,1,0,1,1,1,0]
Phi of -K [-1,-1,0,0,1,1,-1,0,1,1,1,1,1,1,1,1,0,0,0,1,0]
Phi of K* [-1,-1,0,0,1,1,0,0,0,1,1,0,1,1,1,-1,0,1,1,1,1]
Phi of -K* [-1,-1,0,0,1,1,-1,0,0,1,1,0,1,1,1,1,0,1,1,1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+10t^4+8t^2
Outer characteristic polynomial t^7+14t^5+20t^3+3t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7
2-strand cable arrow polynomial -704*K1**6 - 576*K1**4*K2**2 + 3808*K1**4*K2 - 8288*K1**4 + 1568*K1**3*K2*K3 - 1248*K1**3*K3 - 192*K1**2*K2**4 + 1280*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 10576*K1**2*K2**2 - 1184*K1**2*K2*K4 + 14016*K1**2*K2 - 640*K1**2*K3**2 - 112*K1**2*K4**2 - 3804*K1**2 + 608*K1*K2**3*K3 - 1376*K1*K2**2*K3 - 448*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 8336*K1*K2*K3 + 824*K1*K3*K4 + 104*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 1344*K2**4 - 96*K2**3*K6 - 336*K2**2*K3**2 - 128*K2**2*K4**2 + 1416*K2**2*K4 - 4418*K2**2 + 272*K2*K3*K5 + 72*K2*K4*K6 - 1576*K3**2 - 312*K4**2 - 44*K5**2 - 6*K6**2 + 4622
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{6}, {1, 5}, {3, 4}, {2}]]
If K is slice False
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