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Flat knot 6.1177

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,-1,1,1,2,3,0,1,0,0,0,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1177', '7.24088']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.115', '6.407', '6.413', '6.448', '6.844', '6.879', '6.888', '6.926', '6.934', '6.1140', '6.1143', '6.1161', '6.1177']
Outer characteristic polynomial of the knot is: t^7+59t^5+102t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1177', '7.24088']
2-strand cable arrow polynomial of the knot is: -576*K1**4 + 768*K1**3*K2*K3 - 64*K1**3*K3 - 2000*K1**2*K2**2 - 256*K1**2*K2*K4 + 2136*K1**2*K2 - 1088*K1**2*K3**2 - 192*K1**2*K3*K5 - 1324*K1**2 + 960*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 160*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 224*K1*K2*K3*K6 + 3328*K1*K2*K3 - 32*K1*K2*K5*K6 + 1280*K1*K3*K4 + 224*K1*K4*K5 + 32*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4*K4**2 + 192*K2**4*K4 - 864*K2**4 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1248*K2**2*K3**2 - 360*K2**2*K4**2 + 1112*K2**2*K4 - 16*K2**2*K5**2 - 48*K2**2*K6**2 - 1198*K2**2 + 792*K2*K3*K5 + 280*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 1048*K3**2 - 488*K4**2 - 152*K5**2 - 58*K6**2 - 4*K7**2 - 2*K8**2 + 1448
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1177']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.491', 'vk6.563', 'vk6.630', 'vk6.972', 'vk6.1065', 'vk6.1140', 'vk6.1661', 'vk6.1780', 'vk6.1853', 'vk6.2152', 'vk6.2245', 'vk6.2322', 'vk6.2589', 'vk6.2912', 'vk6.3075', 'vk6.3181', 'vk6.12071', 'vk6.13064', 'vk6.20510', 'vk6.21094', 'vk6.21887', 'vk6.22524', 'vk6.27940', 'vk6.28538', 'vk6.29424', 'vk6.32727', 'vk6.39357', 'vk6.41534', 'vk6.46816', 'vk6.46889', 'vk6.53301', 'vk6.57373']
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 O1O2O3O4U1U5U6O5O6U4U3U2
R3 orbit {'O1O2O3O4U1U5U6O5O6U4U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U2U1O5O6U5U6U4
Gauss code of K* O1O2O3U2U3O4O5O6U1U6U5U4
Gauss code of -K* O1O2O3U4U5O4O5O6U3U2U1U6
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 -3 1 1 1 -1 1],[ 3 0 3 2 1 3 3],[-1 -3 0 0 0 -2 0],[-1 -2 0 0 0 -2 0],[-1 -1 0 0 0 -2 0],[ 1 -3 2 2 2 0 1],[-1 -3 0 0 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 0 0 0 -1 -3],[-1 0 0 0 0 -2 -1],[-1 0 0 0 0 -2 -2],[-1 0 0 0 0 -2 -3],[ 1 1 2 2 2 0 -3],[ 3 3 1 2 3 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,1,3,0,0,0,1,3,0,0,2,1,0,2,2,2,3,3]
Phi over symmetry [-3,-1,1,1,1,1,-1,1,1,2,3,0,1,0,0,0,0,0,0,0,0]
Phi of -K [-3,-1,1,1,1,1,-1,1,1,2,3,0,1,0,0,0,0,0,0,0,0]
Phi of K* [-1,-1,-1,-1,1,3,0,0,0,0,1,0,0,0,2,0,0,3,1,1,-1]
Phi of -K* [-3,-1,1,1,1,1,3,1,2,3,3,2,2,1,2,0,0,0,0,0,0]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial z^2+6z+9
Enhanced Jones-Krushkal polynomial -4w^4z^2+5w^3z^2-12w^3z+18w^2z+9w
Inner characteristic polynomial t^6+45t^4+74t^2
Outer characteristic polynomial t^7+59t^5+102t^3+6t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -576*K1**4 + 768*K1**3*K2*K3 - 64*K1**3*K3 - 2000*K1**2*K2**2 - 256*K1**2*K2*K4 + 2136*K1**2*K2 - 1088*K1**2*K3**2 - 192*K1**2*K3*K5 - 1324*K1**2 + 960*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 160*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 224*K1*K2*K3*K6 + 3328*K1*K2*K3 - 32*K1*K2*K5*K6 + 1280*K1*K3*K4 + 224*K1*K4*K5 + 32*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4*K4**2 + 192*K2**4*K4 - 864*K2**4 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1248*K2**2*K3**2 - 360*K2**2*K4**2 + 1112*K2**2*K4 - 16*K2**2*K5**2 - 48*K2**2*K6**2 - 1198*K2**2 + 792*K2*K3*K5 + 280*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 1048*K3**2 - 488*K4**2 - 152*K5**2 - 58*K6**2 - 4*K7**2 - 2*K8**2 + 1448
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}]]
If K is slice False
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