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Flat knot 6.562

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,1,0,2,1,1,1,2,2,1,1,2,-1,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.562']
Arrow polynomial of the knot is: -8*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 5*K2 + 2*K3 + 2*K4 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.562', '6.1915']
Outer characteristic polynomial of the knot is: t^7+41t^5+42t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.562']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 480*K1**4*K2 - 2272*K1**4 + 160*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1280*K1**3*K3 + 64*K1**2*K2**3 - 1632*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 416*K1**2*K2*K4 + 6784*K1**2*K2 - 1216*K1**2*K3**2 - 192*K1**2*K3*K5 - 144*K1**2*K4**2 - 5624*K1**2 - 544*K1*K2**2*K3 - 32*K1*K2**2*K5 + 64*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5992*K1*K2*K3 - 128*K1*K3**2*K5 + 2152*K1*K3*K4 + 384*K1*K4*K5 + 96*K1*K5*K6 - 96*K2**4 - 352*K2**2*K3**2 - 48*K2**2*K4**2 + 608*K2**2*K4 - 8*K2**2*K6**2 - 4080*K2**2 - 32*K2*K3*K4*K5 + 576*K2*K3*K5 + 56*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 - 64*K3**4 - 32*K3**2*K4**2 + 136*K3**2*K6 - 2580*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 890*K4**2 - 300*K5**2 - 88*K6**2 - 16*K7**2 - 12*K8**2 + 4532
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.562']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4778', 'vk6.4798', 'vk6.5113', 'vk6.5133', 'vk6.6343', 'vk6.6771', 'vk6.6791', 'vk6.8305', 'vk6.8317', 'vk6.8755', 'vk6.9675', 'vk6.9687', 'vk6.9984', 'vk6.9996', 'vk6.21016', 'vk6.21031', 'vk6.22438', 'vk6.22455', 'vk6.28467', 'vk6.40236', 'vk6.40251', 'vk6.42164', 'vk6.46734', 'vk6.46749', 'vk6.48802', 'vk6.49017', 'vk6.49037', 'vk6.49833', 'vk6.49853', 'vk6.51500', 'vk6.58962', 'vk6.69794']
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 O1O2O3O4O5U4U2U5O6U3U1U6
R3 orbit {'O1O2O3O4O5U4U2U5O6U3U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U5U3O6U1U4U2
Gauss code of K* O1O2O3U4O5O6O4U6U2U5U1U3
Gauss code of -K* O1O2O3U1O4O5O6U4U6U3U5U2
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 -1 -2 0 -1 2 2],[ 1 0 -2 1 -1 2 2],[ 2 2 0 2 0 2 1],[ 0 -1 -2 0 -1 1 1],[ 1 1 0 1 0 1 0],[-2 -2 -2 -1 -1 0 0],[-2 -2 -1 -1 0 0 0]]
Primitive based matrix [[ 0 2 2 0 -1 -1 -2],[-2 0 0 -1 0 -2 -1],[-2 0 0 -1 -1 -2 -2],[ 0 1 1 0 -1 -1 -2],[ 1 0 1 1 0 1 0],[ 1 2 2 1 -1 0 -2],[ 2 1 2 2 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,1,1,2,0,1,0,2,1,1,1,2,2,1,1,2,-1,0,2]
Phi over symmetry [-2,-2,0,1,1,2,0,1,0,2,1,1,1,2,2,1,1,2,-1,0,2]
Phi of -K [-2,-1,-1,0,2,2,-1,1,0,2,3,1,0,1,1,0,2,3,1,1,0]
Phi of K* [-2,-2,0,1,1,2,0,1,1,2,2,1,1,3,3,0,0,0,-1,-1,1]
Phi of -K* [-2,-1,-1,0,2,2,0,2,2,1,2,1,1,0,1,1,2,2,1,1,0]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 17z+35
Enhanced Jones-Krushkal polynomial 17w^2z+35w
Inner characteristic polynomial t^6+27t^4+15t^2
Outer characteristic polynomial t^7+41t^5+42t^3+4t
Flat arrow polynomial -8*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 5*K2 + 2*K3 + 2*K4 + 6
2-strand cable arrow polynomial -128*K1**6 + 480*K1**4*K2 - 2272*K1**4 + 160*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1280*K1**3*K3 + 64*K1**2*K2**3 - 1632*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 416*K1**2*K2*K4 + 6784*K1**2*K2 - 1216*K1**2*K3**2 - 192*K1**2*K3*K5 - 144*K1**2*K4**2 - 5624*K1**2 - 544*K1*K2**2*K3 - 32*K1*K2**2*K5 + 64*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5992*K1*K2*K3 - 128*K1*K3**2*K5 + 2152*K1*K3*K4 + 384*K1*K4*K5 + 96*K1*K5*K6 - 96*K2**4 - 352*K2**2*K3**2 - 48*K2**2*K4**2 + 608*K2**2*K4 - 8*K2**2*K6**2 - 4080*K2**2 - 32*K2*K3*K4*K5 + 576*K2*K3*K5 + 56*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 - 64*K3**4 - 32*K3**2*K4**2 + 136*K3**2*K6 - 2580*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 890*K4**2 - 300*K5**2 - 88*K6**2 - 16*K7**2 - 12*K8**2 + 4532
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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