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Flat knot 6.1221

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,0,2,3,3,1,3,2,2,1,0,0,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1221']
Arrow polynomial of the knot is: -6*K1**2 - 4*K1*K2 - 4*K1*K3 + 2*K1 + 5*K2 + 2*K3 + 2*K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1221']
Outer characteristic polynomial of the knot is: t^7+31t^5+65t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1221']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 384*K1**4*K2 - 2272*K1**4 + 320*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1280*K1**3*K3 - 1424*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 384*K1**2*K2*K4 + 6080*K1**2*K2 - 2112*K1**2*K3**2 - 128*K1**2*K3*K5 - 224*K1**2*K4**2 - 64*K1**2*K4*K6 - 5896*K1**2 - 384*K1*K2**2*K3 + 128*K1*K2*K3**3 - 256*K1*K2*K3*K4 + 7576*K1*K2*K3 - 192*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 3008*K1*K3*K4 + 440*K1*K4*K5 + 168*K1*K5*K6 + 32*K1*K6*K7 - 56*K2**4 - 496*K2**2*K3**2 - 16*K2**2*K4**2 + 640*K2**2*K4 - 16*K2**2*K6**2 - 4700*K2**2 + 904*K2*K3*K5 + 176*K2*K4*K6 + 32*K2*K5*K7 + 32*K2*K6*K8 - 192*K3**4 + 280*K3**2*K6 - 3596*K3**2 + 48*K3*K4*K7 - 1250*K4**2 - 452*K5**2 - 228*K6**2 - 48*K7**2 - 12*K8**2 + 5428
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1221']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3647', 'vk6.3744', 'vk6.3937', 'vk6.4034', 'vk6.4480', 'vk6.4575', 'vk6.5866', 'vk6.5993', 'vk6.7140', 'vk6.7317', 'vk6.7410', 'vk6.7919', 'vk6.8038', 'vk6.9353', 'vk6.17917', 'vk6.18012', 'vk6.18768', 'vk6.24456', 'vk6.24891', 'vk6.25354', 'vk6.37507', 'vk6.43891', 'vk6.44238', 'vk6.44543', 'vk6.48271', 'vk6.48336', 'vk6.50062', 'vk6.50176', 'vk6.50564', 'vk6.50627', 'vk6.55876', 'vk6.60730']
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 O1O2O3O4U3U1U4O5O6U5U2U6
R3 orbit {'O1O2O3O4U3U1U4O5O6U5U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3U6O5O6U1U4U2
Gauss code of K* O1O2O3U4U5O4O6O5U2U6U1U3
Gauss code of -K* O1O2O3U1U3O4O5O6U4U6U2U5
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 0 -1 2 -1 2],[ 2 0 2 0 2 0 1],[ 0 -2 0 -1 1 0 2],[ 1 0 1 0 1 0 0],[-2 -2 -1 -1 0 0 0],[ 1 0 0 0 0 0 1],[-2 -1 -2 0 0 -1 0]]
Primitive based matrix [[ 0 2 2 0 -1 -1 -2],[-2 0 0 -1 0 -1 -2],[-2 0 0 -2 -1 0 -1],[ 0 1 2 0 0 -1 -2],[ 1 0 1 0 0 0 0],[ 1 1 0 1 0 0 0],[ 2 2 1 2 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,1,1,2,0,1,0,1,2,2,1,0,1,0,1,2,0,0,0]
Phi over symmetry [-2,-2,0,1,1,2,0,0,2,3,3,1,3,2,2,1,0,0,0,1,1]
Phi of -K [-2,-1,-1,0,2,2,1,1,0,2,3,0,0,2,3,1,3,2,1,0,0]
Phi of K* [-2,-2,0,1,1,2,0,0,2,3,3,1,3,2,2,1,0,0,0,1,1]
Phi of -K* [-2,-1,-1,0,2,2,0,0,2,1,2,0,0,1,0,1,0,1,2,1,0]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 19z+39
Enhanced Jones-Krushkal polynomial 19w^2z+39w
Inner characteristic polynomial t^6+17t^4+26t^2
Outer characteristic polynomial t^7+31t^5+65t^3+5t
Flat arrow polynomial -6*K1**2 - 4*K1*K2 - 4*K1*K3 + 2*K1 + 5*K2 + 2*K3 + 2*K4 + 4
2-strand cable arrow polynomial -128*K1**6 + 384*K1**4*K2 - 2272*K1**4 + 320*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1280*K1**3*K3 - 1424*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 384*K1**2*K2*K4 + 6080*K1**2*K2 - 2112*K1**2*K3**2 - 128*K1**2*K3*K5 - 224*K1**2*K4**2 - 64*K1**2*K4*K6 - 5896*K1**2 - 384*K1*K2**2*K3 + 128*K1*K2*K3**3 - 256*K1*K2*K3*K4 + 7576*K1*K2*K3 - 192*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 3008*K1*K3*K4 + 440*K1*K4*K5 + 168*K1*K5*K6 + 32*K1*K6*K7 - 56*K2**4 - 496*K2**2*K3**2 - 16*K2**2*K4**2 + 640*K2**2*K4 - 16*K2**2*K6**2 - 4700*K2**2 + 904*K2*K3*K5 + 176*K2*K4*K6 + 32*K2*K5*K7 + 32*K2*K6*K8 - 192*K3**4 + 280*K3**2*K6 - 3596*K3**2 + 48*K3*K4*K7 - 1250*K4**2 - 452*K5**2 - 228*K6**2 - 48*K7**2 - 12*K8**2 + 5428
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{6}, {1, 5}, {4}, {2, 3}]]
If K is slice False
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