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Flat knot 6.1117

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,-1,1,2,3,-1,1,1,2,0,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1117']
Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 + 2*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.540', '6.925', '6.1021', '6.1117', '6.1120', '6.1135', '6.1227', '6.1230', '6.1260', '6.1682', '6.1685', '6.1922']
Outer characteristic polynomial of the knot is: t^7+36t^5+41t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1117']
2-strand cable arrow polynomial of the knot is: -2576*K1**4 + 736*K1**3*K2*K3 + 192*K1**3*K3*K4 - 480*K1**3*K3 + 352*K1**2*K2**2*K4 - 2720*K1**2*K2**2 + 96*K1**2*K2*K4**2 - 1216*K1**2*K2*K4 + 5976*K1**2*K2 - 1584*K1**2*K3**2 - 96*K1**2*K3*K5 - 688*K1**2*K4**2 - 3928*K1**2 - 800*K1*K2**2*K3 - 64*K1*K2**2*K5 - 544*K1*K2*K3*K4 + 5784*K1*K2*K3 - 96*K1*K2*K4*K5 + 3312*K1*K3*K4 + 664*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**4 - 64*K2**2*K3**2 - 112*K2**2*K4**2 + 1120*K2**2*K4 - 3524*K2**2 + 216*K2*K3*K5 + 112*K2*K4*K6 - 2340*K3**2 - 1348*K4**2 - 156*K5**2 - 28*K6**2 + 3810
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1117']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11486', 'vk6.11791', 'vk6.12808', 'vk6.13143', 'vk6.17063', 'vk6.17306', 'vk6.20906', 'vk6.21061', 'vk6.22317', 'vk6.22489', 'vk6.23785', 'vk6.28385', 'vk6.31251', 'vk6.31604', 'vk6.32824', 'vk6.35579', 'vk6.36034', 'vk6.40027', 'vk6.40306', 'vk6.42081', 'vk6.43276', 'vk6.46561', 'vk6.46767', 'vk6.48024', 'vk6.52249', 'vk6.53406', 'vk6.57712', 'vk6.57723', 'vk6.58897', 'vk6.59948', 'vk6.64424', 'vk6.69755']
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 O1O2O3O4U3U2O5U1U4O6U5U6
R3 orbit {'O1O2O3O4U3U2O5U1U4O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6O5U1U4O6U3U2
Gauss code of K* O1O2U3O4O5U6O3O6U4U2U1U5
Gauss code of -K* O1O2U1O3O4U2O5O6U3U6U5U4
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 -1 2 1 1],[ 2 0 0 0 3 2 1],[ 1 0 0 0 2 1 0],[ 1 0 0 0 1 1 0],[-2 -3 -2 -1 0 1 1],[-1 -2 -1 -1 -1 0 1],[-1 -1 0 0 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 1 1 -1 -2 -3],[-1 -1 0 1 -1 -1 -2],[-1 -1 -1 0 0 0 -1],[ 1 1 1 0 0 0 0],[ 1 2 1 0 0 0 0],[ 2 3 2 1 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,-1,-1,1,2,3,-1,1,1,2,0,0,1,0,0,0]
Phi over symmetry [-2,-1,-1,1,1,2,-1,-1,1,2,3,-1,1,1,2,0,0,1,0,0,0]
Phi of -K [-2,-1,-1,1,1,2,1,1,1,2,1,0,1,2,1,1,2,2,-1,2,2]
Phi of K* [-2,-1,-1,1,1,2,2,2,1,2,1,-1,2,2,2,1,1,1,0,1,1]
Phi of -K* [-2,-1,-1,1,1,2,0,0,1,2,3,0,0,1,1,0,1,2,-1,-1,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z^2+28z+29
Enhanced Jones-Krushkal polynomial 7w^3z^2+28w^2z+29w
Inner characteristic polynomial t^6+24t^4+15t^2+1
Outer characteristic polynomial t^7+36t^5+41t^3+7t
Flat arrow polynomial -4*K1*K2 + 2*K1 + 2*K3 + 1
2-strand cable arrow polynomial -2576*K1**4 + 736*K1**3*K2*K3 + 192*K1**3*K3*K4 - 480*K1**3*K3 + 352*K1**2*K2**2*K4 - 2720*K1**2*K2**2 + 96*K1**2*K2*K4**2 - 1216*K1**2*K2*K4 + 5976*K1**2*K2 - 1584*K1**2*K3**2 - 96*K1**2*K3*K5 - 688*K1**2*K4**2 - 3928*K1**2 - 800*K1*K2**2*K3 - 64*K1*K2**2*K5 - 544*K1*K2*K3*K4 + 5784*K1*K2*K3 - 96*K1*K2*K4*K5 + 3312*K1*K3*K4 + 664*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**4 - 64*K2**2*K3**2 - 112*K2**2*K4**2 + 1120*K2**2*K4 - 3524*K2**2 + 216*K2*K3*K5 + 112*K2*K4*K6 - 2340*K3**2 - 1348*K4**2 - 156*K5**2 - 28*K6**2 + 3810
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
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