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Flat knot 6.99

Min(phi) over symmetries of the knot is: [-4,-2,-1,1,3,3,0,1,4,3,4,0,2,1,2,2,2,3,2,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.99']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.97', '6.99', '6.287']
Outer characteristic polynomial of the knot is: t^7+106t^5+129t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.99']
2-strand cable arrow polynomial of the knot is: -192*K1**2*K2**4 + 672*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 3248*K1**2*K2**2 - 256*K1**2*K2*K4 + 3464*K1**2*K2 - 128*K1**2*K4**2 - 3016*K1**2 + 480*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 - 224*K1*K2**2*K5 - 416*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4552*K1*K2*K3 - 32*K1*K2*K4*K5 + 944*K1*K3*K4 + 224*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 256*K2**4*K4 - 1552*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 96*K2**3*K6 + 64*K2**2*K3**2*K4 - 1168*K2**2*K3**2 - 64*K2**2*K3*K7 - 352*K2**2*K4**2 + 2104*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 2526*K2**2 - 128*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1176*K2*K3*K5 + 296*K2*K4*K6 + 40*K2*K5*K7 - 64*K3**2*K4**2 + 64*K3**2*K6 - 1672*K3**2 + 48*K3*K4*K7 - 842*K4**2 - 296*K5**2 - 74*K6**2 - 8*K7**2 + 2824
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.99']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17001', 'vk6.17243', 'vk6.20532', 'vk6.21932', 'vk6.23407', 'vk6.23714', 'vk6.27986', 'vk6.29453', 'vk6.35473', 'vk6.35919', 'vk6.39388', 'vk6.41579', 'vk6.42906', 'vk6.43206', 'vk6.45963', 'vk6.47640', 'vk6.55170', 'vk6.55415', 'vk6.57396', 'vk6.58571', 'vk6.59549', 'vk6.59888', 'vk6.62061', 'vk6.63048', 'vk6.64974', 'vk6.65181', 'vk6.66940', 'vk6.67797', 'vk6.68263', 'vk6.68418', 'vk6.69548', 'vk6.70248']
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 O1O2O3O4O5O6U2U6U3U1U5U4
R3 orbit {'O1O2O3O4O5O6U2U6U3U1U5U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U3U2U6U4U1U5
Gauss code of K* O1O2O3O4O5O6U4U1U3U6U5U2
Gauss code of -K* O1O2O3O4O5O6U5U2U1U4U6U3
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 -4 -1 3 3 1],[ 2 0 -2 1 4 3 1],[ 4 2 0 2 4 3 1],[ 1 -1 -2 0 2 1 0],[-3 -4 -4 -2 0 0 0],[-3 -3 -3 -1 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 3 3 1 -1 -2 -4],[-3 0 0 0 -1 -3 -3],[-3 0 0 0 -2 -4 -4],[-1 0 0 0 0 -1 -1],[ 1 1 2 0 0 -1 -2],[ 2 3 4 1 1 0 -2],[ 4 3 4 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-1,1,2,4,0,0,1,3,3,0,2,4,4,0,1,1,1,2,2]
Phi over symmetry [-4,-2,-1,1,3,3,0,1,4,3,4,0,2,1,2,2,2,3,2,2,0]
Phi of -K [-4,-2,-1,1,3,3,0,1,4,3,4,0,2,1,2,2,2,3,2,2,0]
Phi of K* [-3,-3,-1,1,2,4,0,2,2,1,3,2,3,2,4,2,2,4,0,1,0]
Phi of -K* [-4,-2,-1,1,3,3,2,2,1,3,4,1,1,3,4,0,1,2,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-2t^3+t^2
Normalized Jones-Krushkal polynomial 8z^2+25z+19
Enhanced Jones-Krushkal polynomial 8w^3z^2+25w^2z+19w
Inner characteristic polynomial t^6+66t^4+20t^2
Outer characteristic polynomial t^7+106t^5+129t^3+4t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + 2
2-strand cable arrow polynomial -192*K1**2*K2**4 + 672*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 3248*K1**2*K2**2 - 256*K1**2*K2*K4 + 3464*K1**2*K2 - 128*K1**2*K4**2 - 3016*K1**2 + 480*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 - 224*K1*K2**2*K5 - 416*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4552*K1*K2*K3 - 32*K1*K2*K4*K5 + 944*K1*K3*K4 + 224*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 256*K2**4*K4 - 1552*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 96*K2**3*K6 + 64*K2**2*K3**2*K4 - 1168*K2**2*K3**2 - 64*K2**2*K3*K7 - 352*K2**2*K4**2 + 2104*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 2526*K2**2 - 128*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1176*K2*K3*K5 + 296*K2*K4*K6 + 40*K2*K5*K7 - 64*K3**2*K4**2 + 64*K3**2*K6 - 1672*K3**2 + 48*K3*K4*K7 - 842*K4**2 - 296*K5**2 - 74*K6**2 - 8*K7**2 + 2824
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}]]
If K is slice False
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