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Flat knot 6.907

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,2,1,3,1,1,1,1,2,2,2,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.907']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 4*K1*K3 + 4*K2 + 2*K3 + K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.364', '6.895', '6.907']
Outer characteristic polynomial of the knot is: t^7+64t^5+104t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.907']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 384*K1**4*K2 - 1264*K1**4 + 384*K1**3*K2*K3 - 416*K1**3*K3 - 256*K1**2*K2**4 + 480*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 3504*K1**2*K2**2 + 32*K1**2*K2*K3*K5 - 384*K1**2*K2*K4 + 5320*K1**2*K2 - 432*K1**2*K3**2 - 16*K1**2*K4**2 - 32*K1**2*K5**2 - 4240*K1**2 + 672*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 - 416*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5616*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1176*K1*K3*K4 + 360*K1*K4*K5 + 80*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 704*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 800*K2**2*K3**2 - 32*K2**2*K3*K7 - 312*K2**2*K4**2 - 32*K2**2*K4*K8 + 1448*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 3636*K2**2 - 96*K2*K3**2*K4 + 1088*K2*K3*K5 + 288*K2*K4*K6 + 56*K2*K5*K7 + 16*K2*K6*K8 + 56*K3**2*K6 - 2132*K3**2 - 852*K4**2 - 388*K5**2 - 68*K6**2 - 8*K7**2 - 2*K8**2 + 3804
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.907']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11232', 'vk6.11311', 'vk6.12497', 'vk6.12608', 'vk6.18217', 'vk6.18553', 'vk6.24683', 'vk6.25103', 'vk6.30910', 'vk6.31033', 'vk6.32098', 'vk6.32217', 'vk6.36811', 'vk6.37271', 'vk6.44055', 'vk6.44395', 'vk6.51994', 'vk6.52089', 'vk6.52875', 'vk6.52922', 'vk6.56011', 'vk6.56285', 'vk6.60553', 'vk6.60895', 'vk6.63649', 'vk6.63694', 'vk6.64081', 'vk6.64126', 'vk6.65676', 'vk6.65964', 'vk6.68724', 'vk6.68933']
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 O1O2O3O4U2U5O6O5U3U1U6U4
R3 orbit {'O1O2O3O4U2U5O6O5U3U1U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U4U2O6O5U6U3
Gauss code of K* O1O2O3O4U2U5U1U4O5O6U3U6
Gauss code of -K* O1O2O3O4U5U2O5O6U1U4U6U3
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 -1 3 1 1],[ 2 0 -1 1 4 2 1],[ 2 1 0 1 2 2 1],[ 1 -1 -1 0 2 1 0],[-3 -4 -2 -2 0 -2 -1],[-1 -2 -2 -1 2 0 1],[-1 -1 -1 0 1 -1 0]]
Primitive based matrix [[ 0 3 1 1 -1 -2 -2],[-3 0 -1 -2 -2 -2 -4],[-1 1 0 -1 0 -1 -1],[-1 2 1 0 -1 -2 -2],[ 1 2 0 1 0 -1 -1],[ 2 2 1 2 1 0 1],[ 2 4 1 2 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,2,2,1,2,2,2,4,1,0,1,1,1,2,2,1,1,-1]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,2,1,3,1,1,1,1,2,2,2,0,0,-1]
Phi of -K [-2,-2,-1,1,1,3,-1,0,1,2,3,0,1,2,1,1,2,2,-1,0,1]
Phi of K* [-3,-1,-1,1,2,2,0,1,2,1,3,1,1,1,1,2,2,2,0,0,-1]
Phi of -K* [-2,-2,-1,1,1,3,-1,1,1,2,4,1,1,2,2,0,1,2,-1,1,2]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2+19w^2z+31w
Inner characteristic polynomial t^6+44t^4+50t^2+1
Outer characteristic polynomial t^7+64t^5+104t^3+5t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 4*K1*K3 + 4*K2 + 2*K3 + K4 + 4
2-strand cable arrow polynomial -192*K1**4*K2**2 + 384*K1**4*K2 - 1264*K1**4 + 384*K1**3*K2*K3 - 416*K1**3*K3 - 256*K1**2*K2**4 + 480*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 3504*K1**2*K2**2 + 32*K1**2*K2*K3*K5 - 384*K1**2*K2*K4 + 5320*K1**2*K2 - 432*K1**2*K3**2 - 16*K1**2*K4**2 - 32*K1**2*K5**2 - 4240*K1**2 + 672*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 - 416*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5616*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1176*K1*K3*K4 + 360*K1*K4*K5 + 80*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 704*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 800*K2**2*K3**2 - 32*K2**2*K3*K7 - 312*K2**2*K4**2 - 32*K2**2*K4*K8 + 1448*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 3636*K2**2 - 96*K2*K3**2*K4 + 1088*K2*K3*K5 + 288*K2*K4*K6 + 56*K2*K5*K7 + 16*K2*K6*K8 + 56*K3**2*K6 - 2132*K3**2 - 852*K4**2 - 388*K5**2 - 68*K6**2 - 8*K7**2 - 2*K8**2 + 3804
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}]]
If K is slice False
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