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Flat knot 6.11

Min(phi) over symmetries of the knot is: [-5,-3,1,2,2,3,1,3,2,5,4,2,1,4,3,0,1,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.11']
Arrow polynomial of the knot is: -8*K1**3*K2 + 16*K1**3 + 4*K1*K2**2 - 6*K1*K2 - 7*K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.11']
Outer characteristic polynomial of the knot is: t^7+140t^5+221t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.11']
2-strand cable arrow polynomial of the knot is: -512*K1**2*K2**4 + 384*K1**2*K2**3 - 2304*K1**2*K2**2 + 1312*K1**2*K2 - 64*K1**2*K4**2 - 1048*K1**2 + 704*K1*K2**3*K3 + 2080*K1*K2*K3 + 336*K1*K3*K4 + 144*K1*K4*K5 - 128*K2**6*K4**2 + 512*K2**6*K4 - 1216*K2**6 + 128*K2**4*K4**3 - 576*K2**4*K4**2 + 1600*K2**4*K4 - 2208*K2**4 + 64*K2**3*K3*K5 + 64*K2**3*K4*K6 - 448*K2**2*K3**2 - 32*K2**2*K4**4 + 128*K2**2*K4**3 - 952*K2**2*K4**2 + 1640*K2**2*K4 - 64*K2**2*K5**2 + 58*K2**2 + 272*K2*K3*K5 + 232*K2*K4*K6 + 16*K2*K5*K7 - 736*K3**2 - 16*K4**4 - 572*K4**2 - 136*K5**2 - 2*K6**2 + 1338
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.11']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.74001', 'vk6.74005', 'vk6.74527', 'vk6.74530', 'vk6.75998', 'vk6.76740', 'vk6.76746', 'vk6.78975', 'vk6.79525', 'vk6.79532', 'vk6.80969', 'vk6.80973', 'vk6.83027', 'vk6.83636', 'vk6.83960', 'vk6.85201', 'vk6.85205', 'vk6.85284', 'vk6.85288', 'vk6.86560', 'vk6.87482', 'vk6.89295', 'vk6.89299', 'vk6.89824']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3O4O5O6U1U2U6U4U5U3
R3 orbit {'O1O2O3O4O5O6U1U2U6U4U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U4U2U3U1U5U6
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U4U2U3U1U5U6
Diagrammatic symmetry type +
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -5 -3 2 1 3 2],[ 5 0 1 5 3 4 2],[ 3 -1 0 4 2 3 1],[-2 -5 -4 0 -1 1 0],[-1 -3 -2 1 0 1 0],[-3 -4 -3 -1 -1 0 0],[-2 -2 -1 0 0 0 0]]
Primitive based matrix [[ 0 3 2 2 1 -3 -5],[-3 0 0 -1 -1 -3 -4],[-2 0 0 0 0 -1 -2],[-2 1 0 0 -1 -4 -5],[-1 1 0 1 0 -2 -3],[ 3 3 1 4 2 0 -1],[ 5 4 2 5 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-2,-1,3,5,0,1,1,3,4,0,0,1,2,1,4,5,2,3,1]
Phi over symmetry [-5,-3,1,2,2,3,1,3,2,5,4,2,1,4,3,0,1,1,0,0,1]
Phi of -K [-5,-3,1,2,2,3,1,3,2,5,4,2,1,4,3,0,1,1,0,0,1]
Phi of K* [-3,-2,-2,-1,3,5,0,1,1,3,4,0,0,1,2,1,4,5,2,3,1]
Phi of -K* [-5,-3,1,2,2,3,1,3,2,5,4,2,1,4,3,0,1,1,0,0,1]
Symmetry type of based matrix +
u-polynomial t^5-2t^2-t
Normalized Jones-Krushkal polynomial z+3
Enhanced Jones-Krushkal polynomial 4w^4z-16w^3z+4w^3+13w^2z-w
Inner characteristic polynomial t^6+88t^4+39t^2
Outer characteristic polynomial t^7+140t^5+221t^3
Flat arrow polynomial -8*K1**3*K2 + 16*K1**3 + 4*K1*K2**2 - 6*K1*K2 - 7*K1 + K3 + 1
2-strand cable arrow polynomial -512*K1**2*K2**4 + 384*K1**2*K2**3 - 2304*K1**2*K2**2 + 1312*K1**2*K2 - 64*K1**2*K4**2 - 1048*K1**2 + 704*K1*K2**3*K3 + 2080*K1*K2*K3 + 336*K1*K3*K4 + 144*K1*K4*K5 - 128*K2**6*K4**2 + 512*K2**6*K4 - 1216*K2**6 + 128*K2**4*K4**3 - 576*K2**4*K4**2 + 1600*K2**4*K4 - 2208*K2**4 + 64*K2**3*K3*K5 + 64*K2**3*K4*K6 - 448*K2**2*K3**2 - 32*K2**2*K4**4 + 128*K2**2*K4**3 - 952*K2**2*K4**2 + 1640*K2**2*K4 - 64*K2**2*K5**2 + 58*K2**2 + 272*K2*K3*K5 + 232*K2*K4*K6 + 16*K2*K5*K7 - 736*K3**2 - 16*K4**4 - 572*K4**2 - 136*K5**2 - 2*K6**2 + 1338
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{3, 6}, {5}, {4}, {2}, {1}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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