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Flat knot 6.261

Min(phi) over symmetries of the knot is: [-4,-2,-1,2,2,3,0,2,3,4,3,1,2,2,1,2,2,2,-1,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.261']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.105', '6.144', '6.261', '6.285', '6.392', '6.480']
Outer characteristic polynomial of the knot is: t^7+105t^5+64t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.261']
2-strand cable arrow polynomial of the knot is: -1056*K1**4 + 608*K1**3*K2*K3 - 800*K1**3*K3 + 64*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 2784*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 6248*K1**2*K2 - 1920*K1**2*K3**2 - 6552*K1**2 + 448*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1024*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 320*K1*K2*K3*K4 + 8064*K1*K2*K3 - 160*K1*K2*K4*K5 + 3168*K1*K3*K4 + 232*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 720*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 800*K2**2*K3**2 - 312*K2**2*K4**2 + 1384*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 4634*K2**2 - 96*K2*K3**2*K4 + 680*K2*K3*K5 + 312*K2*K4*K6 + 8*K2*K5*K7 + 8*K3**2*K6 - 3424*K3**2 - 1422*K4**2 - 240*K5**2 - 70*K6**2 + 5372
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.261']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73328', 'vk6.73342', 'vk6.73489', 'vk6.73504', 'vk6.75250', 'vk6.75269', 'vk6.75498', 'vk6.75514', 'vk6.78218', 'vk6.78226', 'vk6.78456', 'vk6.78469', 'vk6.80044', 'vk6.80053', 'vk6.80192', 'vk6.80202', 'vk6.81932', 'vk6.81933', 'vk6.82196', 'vk6.82214', 'vk6.82656', 'vk6.82660', 'vk6.84716', 'vk6.84722', 'vk6.85020', 'vk6.85022', 'vk6.85750', 'vk6.86505', 'vk6.87325', 'vk6.87687', 'vk6.89627', 'vk6.90078']
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 O1O2O3O4O5U1U3O6U2U5U6U4
R3 orbit {'O1O2O3O4O5U1U3O6U2U5U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U6U1U4O6U3U5
Gauss code of K* O1O2O3O4U5U1U6U4U2O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U3U1U5U4U6
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 -4 -2 -1 3 2 2],[ 4 0 2 1 4 3 2],[ 2 -2 0 0 4 2 2],[ 1 -1 0 0 2 1 1],[-3 -4 -4 -2 0 -1 1],[-2 -3 -2 -1 1 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 2 2 -1 -2 -4],[-3 0 1 -1 -2 -4 -4],[-2 -1 0 -1 -1 -2 -2],[-2 1 1 0 -1 -2 -3],[ 1 2 1 1 0 0 -1],[ 2 4 2 2 0 0 -2],[ 4 4 2 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-2,1,2,4,-1,1,2,4,4,1,1,2,2,1,2,3,0,1,2]
Phi over symmetry [-4,-2,-1,2,2,3,0,2,3,4,3,1,2,2,1,2,2,2,-1,0,2]
Phi of -K [-4,-2,-1,2,2,3,0,2,3,4,3,1,2,2,1,2,2,2,-1,0,2]
Phi of K* [-3,-2,-2,1,2,4,0,2,2,1,3,1,2,2,3,2,2,4,1,2,0]
Phi of -K* [-4,-2,-1,2,2,3,2,1,2,3,4,0,2,2,4,1,1,2,-1,-1,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+67t^4+11t^2
Outer characteristic polynomial t^7+105t^5+64t^3+5t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + 2
2-strand cable arrow polynomial -1056*K1**4 + 608*K1**3*K2*K3 - 800*K1**3*K3 + 64*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 2784*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 6248*K1**2*K2 - 1920*K1**2*K3**2 - 6552*K1**2 + 448*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1024*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 320*K1*K2*K3*K4 + 8064*K1*K2*K3 - 160*K1*K2*K4*K5 + 3168*K1*K3*K4 + 232*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 720*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 800*K2**2*K3**2 - 312*K2**2*K4**2 + 1384*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 4634*K2**2 - 96*K2*K3**2*K4 + 680*K2*K3*K5 + 312*K2*K4*K6 + 8*K2*K5*K7 + 8*K3**2*K6 - 3424*K3**2 - 1422*K4**2 - 240*K5**2 - 70*K6**2 + 5372
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}]]
If K is slice False
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