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Flat knot 6.187

Min(phi) over symmetries of the knot is: [-4,-3,0,2,2,3,0,2,3,5,3,1,2,3,2,1,2,2,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.187']
Arrow polynomial of the knot is: 12*K1**3 + 4*K1**2*K2 - 10*K1**2 - 8*K1*K2 - 2*K1*K3 - 5*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.187']
Outer characteristic polynomial of the knot is: t^7+121t^5+109t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.187']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 416*K1**4*K2 - 1104*K1**4 + 64*K1**3*K2*K3 - 192*K1**3*K3 - 256*K1**2*K2**4 + 1312*K1**2*K2**3 - 5264*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 6704*K1**2*K2 - 144*K1**2*K3**2 - 16*K1**2*K4**2 - 4908*K1**2 + 1280*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 384*K1*K2**2*K5 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6016*K1*K2*K3 + 808*K1*K3*K4 + 80*K1*K4*K5 + 8*K1*K5*K6 - 224*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 352*K2**4*K4 - 2088*K2**4 + 192*K2**3*K3*K5 + 32*K2**3*K4*K6 - 128*K2**3*K6 - 1056*K2**2*K3**2 - 32*K2**2*K3*K7 - 256*K2**2*K4**2 + 2112*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 3026*K2**2 - 32*K2*K3**2*K4 + 544*K2*K3*K5 + 112*K2*K4*K6 + 8*K2*K5*K7 - 1696*K3**2 - 568*K4**2 - 76*K5**2 - 14*K6**2 + 3734
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.187']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73711', 'vk6.73828', 'vk6.74191', 'vk6.74801', 'vk6.75640', 'vk6.75824', 'vk6.76350', 'vk6.76868', 'vk6.78619', 'vk6.78812', 'vk6.79220', 'vk6.79689', 'vk6.80253', 'vk6.80389', 'vk6.80691', 'vk6.81063', 'vk6.81616', 'vk6.81798', 'vk6.81921', 'vk6.82165', 'vk6.82303', 'vk6.82646', 'vk6.83203', 'vk6.84054', 'vk6.84224', 'vk6.84685', 'vk6.85006', 'vk6.86015', 'vk6.87760', 'vk6.88214', 'vk6.89396', 'vk6.89600']
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 O1O2O3O4O5U2O6U1U4U5U6U3
R3 orbit {'O1O2O3O4O5U2O6U1U4U5U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U6U1U2U5O6U4
Gauss code of K* O1O2O3O4O5U1U6U5U2U3O6U4
Gauss code of -K* O1O2O3O4O5U2O6U3U4U1U6U5
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 -3 2 0 2 3],[ 4 0 0 5 2 3 3],[ 3 0 0 3 1 2 2],[-2 -5 -3 0 -2 0 2],[ 0 -2 -1 2 0 1 2],[-2 -3 -2 0 -1 0 1],[-3 -3 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 3 2 2 0 -3 -4],[-3 0 -1 -2 -2 -2 -3],[-2 1 0 0 -1 -2 -3],[-2 2 0 0 -2 -3 -5],[ 0 2 1 2 0 -1 -2],[ 3 2 2 3 1 0 0],[ 4 3 3 5 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-2,0,3,4,1,2,2,2,3,0,1,2,3,2,3,5,1,2,0]
Phi over symmetry [-4,-3,0,2,2,3,0,2,3,5,3,1,2,3,2,1,2,2,0,1,2]
Phi of -K [-4,-3,0,2,2,3,1,2,1,3,4,2,2,3,4,0,1,1,0,-1,0]
Phi of K* [-3,-2,-2,0,3,4,-1,0,1,4,4,0,0,2,1,1,3,3,2,2,1]
Phi of -K* [-4,-3,0,2,2,3,0,2,3,5,3,1,2,3,2,1,2,2,0,1,2]
Symmetry type of based matrix c
u-polynomial t^4-2t^2
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2+21w^2z+27w
Inner characteristic polynomial t^6+79t^4+11t^2
Outer characteristic polynomial t^7+121t^5+109t^3+4t
Flat arrow polynomial 12*K1**3 + 4*K1**2*K2 - 10*K1**2 - 8*K1*K2 - 2*K1*K3 - 5*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -192*K1**4*K2**2 + 416*K1**4*K2 - 1104*K1**4 + 64*K1**3*K2*K3 - 192*K1**3*K3 - 256*K1**2*K2**4 + 1312*K1**2*K2**3 - 5264*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 6704*K1**2*K2 - 144*K1**2*K3**2 - 16*K1**2*K4**2 - 4908*K1**2 + 1280*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 384*K1*K2**2*K5 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6016*K1*K2*K3 + 808*K1*K3*K4 + 80*K1*K4*K5 + 8*K1*K5*K6 - 224*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 352*K2**4*K4 - 2088*K2**4 + 192*K2**3*K3*K5 + 32*K2**3*K4*K6 - 128*K2**3*K6 - 1056*K2**2*K3**2 - 32*K2**2*K3*K7 - 256*K2**2*K4**2 + 2112*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 3026*K2**2 - 32*K2*K3**2*K4 + 544*K2*K3*K5 + 112*K2*K4*K6 + 8*K2*K5*K7 - 1696*K3**2 - 568*K4**2 - 76*K5**2 - 14*K6**2 + 3734
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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