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Flat knot 6.1909

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,0,1,2,1,1,1,1,-1,-1,1,0,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.1909']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 8*K1**2*K2 - 2*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.436', '6.1909']
Outer characteristic polynomial of the knot is: t^7+27t^5+98t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1909']
2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 128*K1**4*K2 - 144*K1**4 + 64*K1**3*K2*K3 - 96*K1**3*K3 + 256*K1**2*K2**5 - 1920*K1**2*K2**4 + 4256*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7984*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 5712*K1**2*K2 - 80*K1**2*K3**2 - 3736*K1**2 + 128*K1*K2**5*K3 - 640*K1*K2**4*K3 - 128*K1*K2**4*K5 + 3744*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2784*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 5992*K1*K2*K3 + 400*K1*K3*K4 + 32*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1760*K2**6 - 128*K2**5*K6 - 128*K2**4*K3**2 - 64*K2**4*K4**2 + 1920*K2**4*K4 - 4952*K2**4 + 160*K2**3*K3*K5 + 64*K2**3*K4*K6 - 256*K2**3*K6 - 1584*K2**2*K3**2 - 32*K2**2*K3*K7 - 368*K2**2*K4**2 + 3352*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 354*K2**2 + 416*K2*K3*K5 + 80*K2*K4*K6 + 8*K2*K5*K7 - 1300*K3**2 - 384*K4**2 - 36*K5**2 - 6*K6**2 + 2734
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1909']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70464', 'vk6.70479', 'vk6.70521', 'vk6.70594', 'vk6.70638', 'vk6.70661', 'vk6.70750', 'vk6.70838', 'vk6.70921', 'vk6.70950', 'vk6.70998', 'vk6.71099', 'vk6.71158', 'vk6.71173', 'vk6.71233', 'vk6.71294', 'vk6.71319', 'vk6.71334', 'vk6.73553', 'vk6.74353', 'vk6.74997', 'vk6.75314', 'vk6.76570', 'vk6.76640', 'vk6.76988', 'vk6.78296', 'vk6.79389', 'vk6.79936', 'vk6.81498', 'vk6.86867', 'vk6.88057', 'vk6.89223']
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 O1O2O3U1U2U4O5O6O4U3U5U6
R3 orbit {'O1O2O3U1U2U4O5O6O4U3U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U5O6O4O5U2U3U6
Gauss code of K* O1O2O3U4U5U1O4O5O6U2U3U6
Gauss code of -K* O1O2O3U4U1U2O4O5O6U3U5U6
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 0 0 2 -1 1],[ 2 0 1 2 2 1 1],[ 0 -1 0 1 0 1 1],[ 0 -2 -1 0 0 0 1],[-2 -2 0 0 0 -1 0],[ 1 -1 -1 0 1 0 1],[-1 -1 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 0 0 -1 -2],[-1 0 0 -1 -1 -1 -1],[ 0 0 1 0 1 1 -1],[ 0 0 1 -1 0 0 -2],[ 1 1 1 -1 0 0 -1],[ 2 2 1 1 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,0,0,1,2,1,1,1,1,-1,-1,1,0,2,1]
Phi over symmetry [-2,-1,0,0,1,2,0,0,0,1,2,1,1,1,1,-1,-1,1,0,2,1]
Phi of -K [-2,-1,0,0,1,2,0,0,1,2,2,1,2,1,2,1,0,2,0,2,1]
Phi of K* [-2,-1,0,0,1,2,1,2,2,2,2,0,0,1,2,-1,1,0,2,1,0]
Phi of -K* [-2,-1,0,0,1,2,1,1,2,1,2,-1,0,1,1,1,1,0,1,0,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+18z+17
Enhanced Jones-Krushkal polynomial -4w^4z^2+9w^3z^2-4w^3z+22w^2z+17w
Inner characteristic polynomial t^6+17t^4+32t^2
Outer characteristic polynomial t^7+27t^5+98t^3+11t
Flat arrow polynomial -8*K1**4 + 4*K1**3 + 8*K1**2*K2 - 2*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -64*K1**4*K2**2 + 128*K1**4*K2 - 144*K1**4 + 64*K1**3*K2*K3 - 96*K1**3*K3 + 256*K1**2*K2**5 - 1920*K1**2*K2**4 + 4256*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7984*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 5712*K1**2*K2 - 80*K1**2*K3**2 - 3736*K1**2 + 128*K1*K2**5*K3 - 640*K1*K2**4*K3 - 128*K1*K2**4*K5 + 3744*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2784*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 5992*K1*K2*K3 + 400*K1*K3*K4 + 32*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1760*K2**6 - 128*K2**5*K6 - 128*K2**4*K3**2 - 64*K2**4*K4**2 + 1920*K2**4*K4 - 4952*K2**4 + 160*K2**3*K3*K5 + 64*K2**3*K4*K6 - 256*K2**3*K6 - 1584*K2**2*K3**2 - 32*K2**2*K3*K7 - 368*K2**2*K4**2 + 3352*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 354*K2**2 + 416*K2*K3*K5 + 80*K2*K4*K6 + 8*K2*K5*K7 - 1300*K3**2 - 384*K4**2 - 36*K5**2 - 6*K6**2 + 2734
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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