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Flat knot 6.553

Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,-1,-1,1,2,3,0,0,2,2,0,1,1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.553']
Arrow polynomial of the knot is: -6*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 4*K2 + 2*K3 + 2*K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.553']
Outer characteristic polynomial of the knot is: t^7+65t^5+64t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.553']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 608*K1**4*K2 - 3072*K1**4 + 352*K1**3*K2*K3 + 160*K1**3*K3*K4 - 1216*K1**3*K3 + 64*K1**2*K2**3 - 2608*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 864*K1**2*K2*K4 + 8008*K1**2*K2 - 1600*K1**2*K3**2 - 64*K1**2*K3*K5 - 400*K1**2*K4**2 - 64*K1**2*K4*K6 - 6280*K1**2 - 448*K1*K2**2*K3 + 32*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7160*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 32*K1*K3**2*K5 - 64*K1*K3*K4*K6 + 2984*K1*K3*K4 + 696*K1*K4*K5 + 160*K1*K5*K6 + 64*K1*K6*K7 - 56*K2**4 - 208*K2**2*K3**2 - 16*K2**2*K4**2 + 752*K2**2*K4 - 8*K2**2*K6**2 - 4760*K2**2 + 512*K2*K3*K5 - 32*K2*K4**2*K6 + 240*K2*K4*K6 + 8*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 - 32*K3**2*K4**2 + 136*K3**2*K6 - 3064*K3**2 + 104*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1352*K4**2 - 368*K5**2 - 200*K6**2 - 56*K7**2 - 4*K8**2 + 5410
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.553']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3663', 'vk6.3760', 'vk6.3953', 'vk6.4050', 'vk6.4482', 'vk6.4577', 'vk6.5864', 'vk6.5991', 'vk6.7156', 'vk6.7333', 'vk6.7426', 'vk6.7921', 'vk6.8040', 'vk6.9351', 'vk6.17901', 'vk6.17996', 'vk6.18770', 'vk6.24440', 'vk6.24889', 'vk6.25352', 'vk6.37509', 'vk6.43875', 'vk6.44236', 'vk6.44541', 'vk6.48287', 'vk6.48352', 'vk6.50078', 'vk6.50192', 'vk6.50566', 'vk6.50629', 'vk6.55860', 'vk6.60728']
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 O1O2O3O4O5U3U6U5O6U1U4U2
R3 orbit {'O1O2O3O4O5U3U6U5O6U1U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U2U5O6U1U6U3
Gauss code of K* O1O2O3U2O4O5O6U4U6U1U5U3
Gauss code of -K* O1O2O3U4O5O4O6U5U2U6U1U3
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 1 -2 2 2 -1],[ 2 0 2 -1 2 2 1],[-1 -2 0 -2 1 2 -2],[ 2 1 2 0 2 1 1],[-2 -2 -1 -2 0 1 -3],[-2 -2 -2 -1 -1 0 -2],[ 1 -1 2 -1 3 2 0]]
Primitive based matrix [[ 0 2 2 1 -1 -2 -2],[-2 0 1 -1 -3 -2 -2],[-2 -1 0 -2 -2 -1 -2],[-1 1 2 0 -2 -2 -2],[ 1 3 2 2 0 -1 -1],[ 2 2 1 2 1 0 1],[ 2 2 2 2 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,2,2,-1,1,3,2,2,2,2,1,2,2,2,2,1,1,-1]
Phi over symmetry [-2,-2,-1,1,2,2,-1,-1,1,2,3,0,0,2,2,0,1,1,0,0,-1]
Phi of -K [-2,-2,-1,1,2,2,-1,0,1,2,3,0,1,2,2,0,0,1,0,-1,-1]
Phi of K* [-2,-2,-1,1,2,2,-1,-1,1,2,3,0,0,2,2,0,1,1,0,0,-1]
Phi of -K* [-2,-2,-1,1,2,2,-1,1,2,2,2,1,2,1,2,2,2,3,2,1,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 19z+39
Enhanced Jones-Krushkal polynomial 19w^2z+39w
Inner characteristic polynomial t^6+47t^4+36t^2+1
Outer characteristic polynomial t^7+65t^5+64t^3+6t
Flat arrow polynomial -6*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 4*K2 + 2*K3 + 2*K4 + 5
2-strand cable arrow polynomial -128*K1**6 + 608*K1**4*K2 - 3072*K1**4 + 352*K1**3*K2*K3 + 160*K1**3*K3*K4 - 1216*K1**3*K3 + 64*K1**2*K2**3 - 2608*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 864*K1**2*K2*K4 + 8008*K1**2*K2 - 1600*K1**2*K3**2 - 64*K1**2*K3*K5 - 400*K1**2*K4**2 - 64*K1**2*K4*K6 - 6280*K1**2 - 448*K1*K2**2*K3 + 32*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7160*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 32*K1*K3**2*K5 - 64*K1*K3*K4*K6 + 2984*K1*K3*K4 + 696*K1*K4*K5 + 160*K1*K5*K6 + 64*K1*K6*K7 - 56*K2**4 - 208*K2**2*K3**2 - 16*K2**2*K4**2 + 752*K2**2*K4 - 8*K2**2*K6**2 - 4760*K2**2 + 512*K2*K3*K5 - 32*K2*K4**2*K6 + 240*K2*K4*K6 + 8*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 - 32*K3**2*K4**2 + 136*K3**2*K6 - 3064*K3**2 + 104*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1352*K4**2 - 368*K5**2 - 200*K6**2 - 56*K7**2 - 4*K8**2 + 5410
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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