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Flat knot 6.449

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,1,3,1,4,4,1,0,1,2,0,1,2,-1,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.449']
Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.62', '6.176', '6.181', '6.194', '6.228', '6.267', '6.268', '6.449']
Outer characteristic polynomial of the knot is: t^7+88t^5+161t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.449']
2-strand cable arrow polynomial of the knot is: -16*K1**4 + 32*K1**3*K2*K3 - 800*K1**3*K3 - 288*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 3176*K1**2*K2 - 912*K1**2*K3**2 - 96*K1**2*K3*K5 - 4296*K1**2 + 96*K1*K2**3*K3 - 864*K1*K2**2*K3 + 32*K1*K2*K3**3 - 352*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 4808*K1*K2*K3 - 32*K1*K2*K5*K6 - 128*K1*K3**2*K5 + 1576*K1*K3*K4 + 168*K1*K4*K5 + 96*K1*K5*K6 + 40*K1*K6*K7 - 72*K2**4 - 32*K2**3*K6 - 608*K2**2*K3**2 - 8*K2**2*K4**2 + 704*K2**2*K4 - 8*K2**2*K6**2 - 3314*K2**2 + 992*K2*K3*K5 + 128*K2*K4*K6 + 48*K2*K5*K7 + 16*K2*K6*K8 - 128*K3**4 + 232*K3**2*K6 - 2432*K3**2 + 8*K3*K4*K7 + 24*K3*K5*K8 - 684*K4**2 - 368*K5**2 - 150*K6**2 - 40*K7**2 - 18*K8**2 + 3436
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.449']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11438', 'vk6.11735', 'vk6.12752', 'vk6.13097', 'vk6.20316', 'vk6.21659', 'vk6.27620', 'vk6.29166', 'vk6.31193', 'vk6.31534', 'vk6.32361', 'vk6.32778', 'vk6.39044', 'vk6.41306', 'vk6.45800', 'vk6.47477', 'vk6.52203', 'vk6.52466', 'vk6.53034', 'vk6.53356', 'vk6.57187', 'vk6.58400', 'vk6.61801', 'vk6.62924', 'vk6.63773', 'vk6.63885', 'vk6.64201', 'vk6.64389', 'vk6.66800', 'vk6.67670', 'vk6.69440', 'vk6.70164']
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 O1O2O3O4O5U6U3O6U1U4U2U5
R3 orbit {'O1O2O3O4O5U6U3O6U1U4U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U4U2U5O6U3U6
Gauss code of K* O1O2O3O4U1U3U5U2U4O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U1U3U6U2U4
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 -3 0 -1 1 4 -1],[ 3 0 2 1 2 4 2],[ 0 -2 0 0 1 3 -1],[ 1 -1 0 0 0 1 1],[-1 -2 -1 0 0 1 -1],[-4 -4 -3 -1 -1 0 -4],[ 1 -2 1 -1 1 4 0]]
Primitive based matrix [[ 0 4 1 0 -1 -1 -3],[-4 0 -1 -3 -1 -4 -4],[-1 1 0 -1 0 -1 -2],[ 0 3 1 0 0 -1 -2],[ 1 1 0 0 0 1 -1],[ 1 4 1 1 -1 0 -2],[ 3 4 2 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,0,1,1,3,1,3,1,4,4,1,0,1,2,0,1,2,-1,1,2]
Phi over symmetry [-4,-1,0,1,1,3,1,3,1,4,4,1,0,1,2,0,1,2,-1,1,2]
Phi of -K [-3,-1,-1,0,1,4,0,1,1,2,3,1,0,1,1,1,2,4,0,1,2]
Phi of K* [-4,-1,0,1,1,3,2,1,1,4,3,0,1,2,2,0,1,1,-1,0,1]
Phi of -K* [-3,-1,-1,0,1,4,1,2,2,2,4,1,0,0,1,1,1,4,1,3,1]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t
Normalized Jones-Krushkal polynomial 3z^2+20z+29
Enhanced Jones-Krushkal polynomial 3w^3z^2+20w^2z+29w
Inner characteristic polynomial t^6+60t^4+95t^2
Outer characteristic polynomial t^7+88t^5+161t^3+4t
Flat arrow polynomial -2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -16*K1**4 + 32*K1**3*K2*K3 - 800*K1**3*K3 - 288*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 3176*K1**2*K2 - 912*K1**2*K3**2 - 96*K1**2*K3*K5 - 4296*K1**2 + 96*K1*K2**3*K3 - 864*K1*K2**2*K3 + 32*K1*K2*K3**3 - 352*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 4808*K1*K2*K3 - 32*K1*K2*K5*K6 - 128*K1*K3**2*K5 + 1576*K1*K3*K4 + 168*K1*K4*K5 + 96*K1*K5*K6 + 40*K1*K6*K7 - 72*K2**4 - 32*K2**3*K6 - 608*K2**2*K3**2 - 8*K2**2*K4**2 + 704*K2**2*K4 - 8*K2**2*K6**2 - 3314*K2**2 + 992*K2*K3*K5 + 128*K2*K4*K6 + 48*K2*K5*K7 + 16*K2*K6*K8 - 128*K3**4 + 232*K3**2*K6 - 2432*K3**2 + 8*K3*K4*K7 + 24*K3*K5*K8 - 684*K4**2 - 368*K5**2 - 150*K6**2 - 40*K7**2 - 18*K8**2 + 3436
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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