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Flat knot 6.205

Min(phi) over symmetries of the knot is: [-3,-3,-1,2,2,3,-1,1,2,4,4,1,1,3,2,1,2,2,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.205']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 6*K1*K2 - 3*K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.205', '6.660', '6.775', '6.820']
Outer characteristic polynomial of the knot is: t^7+100t^5+52t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.205']
2-strand cable arrow polynomial of the knot is: -96*K1**4 + 128*K1**2*K2**3 - 768*K1**2*K2**2 + 1024*K1**2*K2 - 32*K1**2*K3**2 - 64*K1**2*K4**2 - 920*K1**2 + 64*K1*K2**3*K3 + 688*K1*K2*K3 + 240*K1*K3*K4 + 64*K1*K4*K5 + 16*K1*K5*K6 - 64*K2**6 + 64*K2**4*K4 - 528*K2**4 - 64*K2**2*K3**2 - 88*K2**2*K4**2 + 440*K2**2*K4 - 406*K2**2 + 64*K2*K3*K5 + 56*K2*K4*K6 - 296*K3**2 - 216*K4**2 - 48*K5**2 - 18*K6**2 + 758
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.205']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71613', 'vk6.71737', 'vk6.72154', 'vk6.72342', 'vk6.73764', 'vk6.73793', 'vk6.73901', 'vk6.75737', 'vk6.75903', 'vk6.75920', 'vk6.77229', 'vk6.77537', 'vk6.78703', 'vk6.78718', 'vk6.78746', 'vk6.78901', 'vk6.78921', 'vk6.80325', 'vk6.80336', 'vk6.80350', 'vk6.80447', 'vk6.80459', 'vk6.81723', 'vk6.84461', 'vk6.87991', 'vk6.88345', 'vk6.88356', 'vk6.89316']
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 O1O2O3O4O5U2O6U4U1U6U5U3
R3 orbit {'O1O2O3O4O5U2U6U5U1U4O6U3', 'O1O2O3O4O5U2O6U4U1U6U5U3', 'O1O2O3O4U1O5U6U2U5U4O6U3'}
R3 orbit length 3
Gauss code of -K O1O2O3O4O5U3U1U6U5U2O6U4
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5U3O6U2U5U1U6U4
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 -3 -3 2 -1 3 2],[ 3 0 -1 4 1 4 2],[ 3 1 0 3 1 2 1],[-2 -4 -3 0 -2 1 1],[ 1 -1 -1 2 0 2 1],[-3 -4 -2 -1 -2 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 3 2 2 -1 -3 -3],[-3 0 0 -1 -2 -2 -4],[-2 0 0 -1 -1 -1 -2],[-2 1 1 0 -2 -3 -4],[ 1 2 1 2 0 -1 -1],[ 3 2 1 3 1 0 1],[ 3 4 2 4 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-2,1,3,3,0,1,2,2,4,1,1,1,2,2,3,4,1,1,-1]
Phi over symmetry [-3,-3,-1,2,2,3,-1,1,2,4,4,1,1,3,2,1,2,2,-1,0,1]
Phi of -K [-3,-3,-1,2,2,3,-1,1,2,4,4,1,1,3,2,1,2,2,-1,0,1]
Phi of K* [-3,-2,-2,1,3,3,0,1,2,2,4,1,1,1,2,2,3,4,1,1,-1]
Phi of -K* [-3,-3,-1,2,2,3,-1,1,2,4,4,1,1,3,2,1,2,2,-1,0,1]
Symmetry type of based matrix +
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial -4w^3z+9w^2z+11w
Inner characteristic polynomial t^6+64t^4+20t^2
Outer characteristic polynomial t^7+100t^5+52t^3
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 6*K1*K2 - 3*K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -96*K1**4 + 128*K1**2*K2**3 - 768*K1**2*K2**2 + 1024*K1**2*K2 - 32*K1**2*K3**2 - 64*K1**2*K4**2 - 920*K1**2 + 64*K1*K2**3*K3 + 688*K1*K2*K3 + 240*K1*K3*K4 + 64*K1*K4*K5 + 16*K1*K5*K6 - 64*K2**6 + 64*K2**4*K4 - 528*K2**4 - 64*K2**2*K3**2 - 88*K2**2*K4**2 + 440*K2**2*K4 - 406*K2**2 + 64*K2*K3*K5 + 56*K2*K4*K6 - 296*K3**2 - 216*K4**2 - 48*K5**2 - 18*K6**2 + 758
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {4}, {1, 2}]]
If K is slice False
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