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Flat knot 6.665

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,0,1,1,3,1,0,1,2,0,1,1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.665']
Arrow polynomial of the knot is: -14*K1**2 - 4*K1*K2 + 2*K1 + 7*K2 + 2*K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.665', '6.1301', '6.1514', '6.1646', '6.1669', '6.1709', '6.1710', '6.1744', '6.1776']
Outer characteristic polynomial of the knot is: t^7+35t^5+52t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.665']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 1920*K1**4*K2 - 4112*K1**4 + 480*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1184*K1**3*K3 + 512*K1**2*K2**3 - 6368*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 12104*K1**2*K2 - 624*K1**2*K3**2 - 32*K1**2*K3*K5 - 64*K1**2*K4**2 - 8012*K1**2 + 128*K1*K2**3*K3 - 1184*K1*K2**2*K3 - 64*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 8496*K1*K2*K3 + 1504*K1*K3*K4 + 176*K1*K4*K5 - 696*K2**4 - 240*K2**2*K3**2 - 48*K2**2*K4**2 + 1424*K2**2*K4 - 6188*K2**2 + 296*K2*K3*K5 + 32*K2*K4*K6 - 2704*K3**2 - 806*K4**2 - 100*K5**2 - 4*K6**2 + 6284
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.665']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3561', 'vk6.3582', 'vk6.3600', 'vk6.3803', 'vk6.3825', 'vk6.3836', 'vk6.3858', 'vk6.6969', 'vk6.6995', 'vk6.7002', 'vk6.7028', 'vk6.7187', 'vk6.7217', 'vk6.7220', 'vk6.15331', 'vk6.15348', 'vk6.15456', 'vk6.15475', 'vk6.33972', 'vk6.34016', 'vk6.34031', 'vk6.34431', 'vk6.48222', 'vk6.48224', 'vk6.48379', 'vk6.49954', 'vk6.49976', 'vk6.49986', 'vk6.53992', 'vk6.54013', 'vk6.54046', 'vk6.54496']
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 O1O2O3O4U2O5U4U5O6U1U6U3
R3 orbit {'O1O2O3O4U2O5U4U5O6U1U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U4O5U6U1O6U3
Gauss code of K* O1O2O3U1U4U3U5O4U6O5O6U2
Gauss code of -K* O1O2O3U2O4O5U4O6U5U1U6U3
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 -2 2 0 1 1],[ 2 0 -1 3 0 1 1],[ 2 1 0 2 1 1 0],[-2 -3 -2 0 -1 1 0],[ 0 0 -1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 0 -2 -2],[-2 0 1 0 -1 -2 -3],[-1 -1 0 0 -1 -1 -1],[-1 0 0 0 0 0 -1],[ 0 1 1 0 0 -1 0],[ 2 2 1 0 1 0 1],[ 2 3 1 1 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,2,2,-1,0,1,2,3,0,1,1,1,0,0,1,1,0,-1]
Phi over symmetry [-2,-2,0,1,1,2,-1,0,1,1,3,1,0,1,2,0,1,1,0,0,-1]
Phi of -K [-2,-2,0,1,1,2,-1,1,2,3,2,2,2,2,1,0,1,1,0,2,1]
Phi of K* [-2,-1,-1,0,2,2,1,2,1,1,2,0,1,2,3,0,2,2,2,1,-1]
Phi of -K* [-2,-2,0,1,1,2,-1,0,1,1,3,1,0,1,2,0,1,1,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 21z+43
Enhanced Jones-Krushkal polynomial 21w^2z+43w
Inner characteristic polynomial t^6+21t^4+21t^2
Outer characteristic polynomial t^7+35t^5+52t^3+10t
Flat arrow polynomial -14*K1**2 - 4*K1*K2 + 2*K1 + 7*K2 + 2*K3 + 8
2-strand cable arrow polynomial -192*K1**4*K2**2 + 1920*K1**4*K2 - 4112*K1**4 + 480*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1184*K1**3*K3 + 512*K1**2*K2**3 - 6368*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 12104*K1**2*K2 - 624*K1**2*K3**2 - 32*K1**2*K3*K5 - 64*K1**2*K4**2 - 8012*K1**2 + 128*K1*K2**3*K3 - 1184*K1*K2**2*K3 - 64*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 8496*K1*K2*K3 + 1504*K1*K3*K4 + 176*K1*K4*K5 - 696*K2**4 - 240*K2**2*K3**2 - 48*K2**2*K4**2 + 1424*K2**2*K4 - 6188*K2**2 + 296*K2*K3*K5 + 32*K2*K4*K6 - 2704*K3**2 - 806*K4**2 - 100*K5**2 - 4*K6**2 + 6284
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]]
If K is slice False
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