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Flat knot 6.1090

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,0,1,2,3,4,1,0,2,1,-1,0,-1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1090']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.635', '6.639', '6.659', '6.727', '6.732', '6.735', '6.799', '6.1088', '6.1090', '6.1095', '6.1383']
Outer characteristic polynomial of the knot is: t^7+41t^5+59t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1090']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 768*K1**4*K2**2 + 1056*K1**4*K2 - 1440*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 - 448*K1**2*K2**4 + 1024*K1**2*K2**3 - 2704*K1**2*K2**2 + 2744*K1**2*K2 - 128*K1**2*K3**2 - 48*K1**2*K4**2 - 1288*K1**2 + 224*K1*K2**3*K3 + 1552*K1*K2*K3 + 248*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 528*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 120*K2**2*K4 - 696*K2**2 + 8*K2*K3*K5 - 388*K3**2 - 116*K4**2 - 4*K5**2 + 1226
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1090']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11594', 'vk6.11598', 'vk6.11947', 'vk6.11951', 'vk6.12940', 'vk6.12944', 'vk6.13253', 'vk6.20428', 'vk6.20430', 'vk6.21793', 'vk6.27792', 'vk6.27794', 'vk6.29310', 'vk6.31389', 'vk6.31393', 'vk6.32567', 'vk6.32571', 'vk6.32953', 'vk6.39218', 'vk6.39220', 'vk6.41438', 'vk6.47559', 'vk6.53189', 'vk6.53193', 'vk6.53504', 'vk6.57297', 'vk6.57299', 'vk6.61979', 'vk6.61981', 'vk6.64282', 'vk6.64286', 'vk6.64494']
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 O1O2O3O4U1U4O5U2U3O6U5U6
R3 orbit {'O1O2O3O4U1U4O5U2U3O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6O5U2U3O6U1U4
Gauss code of K* O1O2U3O4O5U6O3O6U1U4U5U2
Gauss code of -K* O1O2U1O3O4U2O5O6U5U3U4U6
Diagrammatic symmetry type c
Flat genus of the diagram 2
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 2 3 1 2 0],[ 1 -2 0 1 0 2 1],[-1 -3 -1 0 0 1 1],[-1 -1 0 0 0 0 0],[-1 -2 -2 -1 0 0 1],[-1 0 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 1 1 0 -1 -3],[-1 -1 0 1 0 -2 -2],[-1 -1 -1 0 0 -1 0],[-1 0 0 0 0 0 -1],[ 1 1 2 1 0 0 -2],[ 3 3 2 0 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,1,3,-1,-1,0,1,3,-1,0,2,2,0,1,0,0,1,2]
Phi over symmetry [-3,-1,1,1,1,1,0,1,2,3,4,1,0,2,1,-1,0,-1,0,-1,0]
Phi of -K [-3,-1,1,1,1,1,0,1,2,3,4,1,0,2,1,-1,0,-1,0,-1,0]
Phi of K* [-1,-1,-1,-1,1,3,-1,-1,0,1,4,-1,0,0,2,0,1,1,2,3,0]
Phi of -K* [-3,-1,1,1,1,1,2,0,1,2,3,1,0,2,1,0,-1,-1,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 9z+19
Enhanced Jones-Krushkal polynomial -4w^3z+13w^2z+19w
Inner characteristic polynomial t^6+27t^4+15t^2
Outer characteristic polynomial t^7+41t^5+59t^3
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
2-strand cable arrow polynomial 256*K1**4*K2**3 - 768*K1**4*K2**2 + 1056*K1**4*K2 - 1440*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 - 448*K1**2*K2**4 + 1024*K1**2*K2**3 - 2704*K1**2*K2**2 + 2744*K1**2*K2 - 128*K1**2*K3**2 - 48*K1**2*K4**2 - 1288*K1**2 + 224*K1*K2**3*K3 + 1552*K1*K2*K3 + 248*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 528*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 120*K2**2*K4 - 696*K2**2 + 8*K2*K3*K5 - 388*K3**2 - 116*K4**2 - 4*K5**2 + 1226
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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