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Flat knot 6.965

Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,0,2,2,1,3,0,1,1,1,0,1,2,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.965']
Arrow polynomial of the knot is: 12*K1**3 + 8*K1**2*K2 - 14*K1**2 - 6*K1*K2 - 4*K1*K3 - 6*K1 + 5*K2 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.965']
Outer characteristic polynomial of the knot is: t^7+43t^5+76t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.965']
2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 480*K1**4*K2 - 2784*K1**4 + 128*K1**3*K2**3*K3 + 832*K1**3*K2*K3 - 512*K1**3*K3 - 768*K1**2*K2**4 + 1600*K1**2*K2**3 - 448*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 9440*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 1056*K1**2*K2*K4 + 11264*K1**2*K2 - 544*K1**2*K3**2 - 112*K1**2*K4**2 - 6956*K1**2 + 3328*K1*K2**3*K3 + 544*K1*K2**2*K3*K4 - 1952*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 576*K1*K2**2*K5 + 64*K1*K2*K3**3 - 320*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10272*K1*K2*K3 - 96*K1*K2*K4*K5 + 1464*K1*K3*K4 + 160*K1*K4*K5 - 224*K2**6 - 320*K2**4*K3**2 - 64*K2**4*K4**2 + 288*K2**4*K4 - 2888*K2**4 + 320*K2**3*K3*K5 + 128*K2**3*K4*K6 - 64*K2**3*K6 - 2400*K2**2*K3**2 - 488*K2**2*K4**2 + 2544*K2**2*K4 - 112*K2**2*K5**2 - 48*K2**2*K6**2 - 4248*K2**2 - 64*K2*K3**2*K4 + 944*K2*K3*K5 + 240*K2*K4*K6 - 2752*K3**2 - 806*K4**2 - 92*K5**2 - 24*K6**2 + 5652
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.965']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3618', 'vk6.3693', 'vk6.3884', 'vk6.4001', 'vk6.7040', 'vk6.7081', 'vk6.7256', 'vk6.7371', 'vk6.17698', 'vk6.17747', 'vk6.24245', 'vk6.24306', 'vk6.36540', 'vk6.36617', 'vk6.43646', 'vk6.43753', 'vk6.48254', 'vk6.48333', 'vk6.48416', 'vk6.48433', 'vk6.50010', 'vk6.50051', 'vk6.50134', 'vk6.50153', 'vk6.55722', 'vk6.55779', 'vk6.60294', 'vk6.60377', 'vk6.65430', 'vk6.65459', 'vk6.68558', 'vk6.68587']
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 O1O2O3O4U5U3O6O5U6U1U2U4
R3 orbit {'O1O2O3O4U5U3O6O5U6U1U2U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U3U4U5O6O5U2U6
Gauss code of K* O1O2O3O4U2U3U5U4O6O5U1U6
Gauss code of -K* O1O2O3O4U5U4O6O5U1U6U2U3
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 0 3 0 -1],[ 2 0 1 1 3 2 -1],[ 0 -1 0 1 2 0 -1],[ 0 -1 -1 0 0 0 -1],[-3 -3 -2 0 0 -2 -1],[ 0 -2 0 0 2 0 -1],[ 1 1 1 1 1 1 0]]
Primitive based matrix [[ 0 3 0 0 0 -1 -2],[-3 0 0 -2 -2 -1 -3],[ 0 0 0 0 -1 -1 -1],[ 0 2 0 0 0 -1 -2],[ 0 2 1 0 0 -1 -1],[ 1 1 1 1 1 0 1],[ 2 3 1 2 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,0,0,0,1,2,0,2,2,1,3,0,1,1,1,0,1,2,1,1,-1]
Phi over symmetry [-3,0,0,0,1,2,0,2,2,1,3,0,1,1,1,0,1,2,1,1,-1]
Phi of -K [-2,-1,0,0,0,3,2,0,1,1,2,0,0,0,3,0,0,1,-1,1,3]
Phi of K* [-3,0,0,0,1,2,1,1,3,3,2,0,0,0,0,1,0,1,0,1,2]
Phi of -K* [-2,-1,0,0,0,3,-1,1,1,2,3,1,1,1,1,-1,0,0,0,2,2]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+29t^4+31t^2+1
Outer characteristic polynomial t^7+43t^5+76t^3+8t
Flat arrow polynomial 12*K1**3 + 8*K1**2*K2 - 14*K1**2 - 6*K1*K2 - 4*K1*K3 - 6*K1 + 5*K2 + 6
2-strand cable arrow polynomial -320*K1**4*K2**2 + 480*K1**4*K2 - 2784*K1**4 + 128*K1**3*K2**3*K3 + 832*K1**3*K2*K3 - 512*K1**3*K3 - 768*K1**2*K2**4 + 1600*K1**2*K2**3 - 448*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 9440*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 1056*K1**2*K2*K4 + 11264*K1**2*K2 - 544*K1**2*K3**2 - 112*K1**2*K4**2 - 6956*K1**2 + 3328*K1*K2**3*K3 + 544*K1*K2**2*K3*K4 - 1952*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 576*K1*K2**2*K5 + 64*K1*K2*K3**3 - 320*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10272*K1*K2*K3 - 96*K1*K2*K4*K5 + 1464*K1*K3*K4 + 160*K1*K4*K5 - 224*K2**6 - 320*K2**4*K3**2 - 64*K2**4*K4**2 + 288*K2**4*K4 - 2888*K2**4 + 320*K2**3*K3*K5 + 128*K2**3*K4*K6 - 64*K2**3*K6 - 2400*K2**2*K3**2 - 488*K2**2*K4**2 + 2544*K2**2*K4 - 112*K2**2*K5**2 - 48*K2**2*K6**2 - 4248*K2**2 - 64*K2*K3**2*K4 + 944*K2*K3*K5 + 240*K2*K4*K6 - 2752*K3**2 - 806*K4**2 - 92*K5**2 - 24*K6**2 + 5652
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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