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Flat knot 6.491

Min(phi) over symmetries of the knot is: [-4,0,0,1,1,2,1,2,1,4,3,0,1,1,1,1,1,2,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.491']
Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.83', '6.151', '6.160', '6.190', '6.247', '6.262', '6.491', '6.514']
Outer characteristic polynomial of the knot is: t^7+65t^5+84t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.491']
2-strand cable arrow polynomial of the knot is: 640*K1**4*K2 - 1392*K1**4 + 256*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1120*K1**3*K3 + 384*K1**2*K2**3 - 1968*K1**2*K2**2 - 1248*K1**2*K2*K4 + 5424*K1**2*K2 - 688*K1**2*K3**2 - 320*K1**2*K4**2 - 5448*K1**2 + 320*K1*K2**3*K3 - 928*K1*K2**2*K3 - 96*K1*K2**2*K5 - 544*K1*K2*K3*K4 + 6152*K1*K2*K3 - 64*K1*K2*K4*K5 + 3000*K1*K3*K4 + 544*K1*K4*K5 + 16*K1*K5*K6 - 360*K2**4 - 480*K2**2*K3**2 - 80*K2**2*K4**2 + 1528*K2**2*K4 - 4108*K2**2 - 128*K2*K3**2*K4 + 680*K2*K3*K5 + 176*K2*K4*K6 - 64*K3**4 - 16*K3**2*K4**2 + 128*K3**2*K6 - 2860*K3**2 + 32*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1574*K4**2 - 296*K5**2 - 76*K6**2 - 12*K7**2 - 2*K8**2 + 4494
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.491']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.14109', 'vk6.14114', 'vk6.14322', 'vk6.14333', 'vk6.15551', 'vk6.15564', 'vk6.16031', 'vk6.16038', 'vk6.16434', 'vk6.16443', 'vk6.16457', 'vk6.22842', 'vk6.22852', 'vk6.34059', 'vk6.34118', 'vk6.34455', 'vk6.34495', 'vk6.34788', 'vk6.34807', 'vk6.34829', 'vk6.42405', 'vk6.42423', 'vk6.54074', 'vk6.54079', 'vk6.54296', 'vk6.54307', 'vk6.54671', 'vk6.54690', 'vk6.54704', 'vk6.64528', 'vk6.64537', 'vk6.64739']
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 O1O2O3O4O5U1U4U3O6U5U6U2
R3 orbit {'O1O2O3O4O5U1U4U3O6U5U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U6U1O6U3U2U5
Gauss code of K* O1O2O3U4O5O4O6U1U6U3U2U5
Gauss code of -K* O1O2O3U2O4O5O6U3U5U4U1U6
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 1 0 0 2 1],[ 4 0 4 2 1 3 1],[-1 -4 0 -1 -1 1 1],[ 0 -2 1 0 0 2 1],[ 0 -1 1 0 0 1 1],[-2 -3 -1 -2 -1 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 0 -4],[-2 0 1 -1 -1 -2 -3],[-1 -1 0 -1 -1 -1 -1],[-1 1 1 0 -1 -1 -4],[ 0 1 1 1 0 0 -1],[ 0 2 1 1 0 0 -2],[ 4 3 1 4 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,0,4,-1,1,1,2,3,1,1,1,1,1,1,4,0,1,2]
Phi over symmetry [-4,0,0,1,1,2,1,2,1,4,3,0,1,1,1,1,1,2,-1,-1,1]
Phi of -K [-4,0,0,1,1,2,2,3,1,4,3,0,0,0,0,0,0,1,-1,0,2]
Phi of K* [-2,-1,-1,0,0,4,0,2,0,1,3,1,0,0,1,0,0,4,0,2,3]
Phi of -K* [-4,0,0,1,1,2,1,2,1,4,3,0,1,1,1,1,1,2,-1,-1,1]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+43t^4+25t^2+1
Outer characteristic polynomial t^7+65t^5+84t^3+10t
Flat arrow polynomial -2*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3
2-strand cable arrow polynomial 640*K1**4*K2 - 1392*K1**4 + 256*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1120*K1**3*K3 + 384*K1**2*K2**3 - 1968*K1**2*K2**2 - 1248*K1**2*K2*K4 + 5424*K1**2*K2 - 688*K1**2*K3**2 - 320*K1**2*K4**2 - 5448*K1**2 + 320*K1*K2**3*K3 - 928*K1*K2**2*K3 - 96*K1*K2**2*K5 - 544*K1*K2*K3*K4 + 6152*K1*K2*K3 - 64*K1*K2*K4*K5 + 3000*K1*K3*K4 + 544*K1*K4*K5 + 16*K1*K5*K6 - 360*K2**4 - 480*K2**2*K3**2 - 80*K2**2*K4**2 + 1528*K2**2*K4 - 4108*K2**2 - 128*K2*K3**2*K4 + 680*K2*K3*K5 + 176*K2*K4*K6 - 64*K3**4 - 16*K3**2*K4**2 + 128*K3**2*K6 - 2860*K3**2 + 32*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1574*K4**2 - 296*K5**2 - 76*K6**2 - 12*K7**2 - 2*K8**2 + 4494
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}]]
If K is slice False
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