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Flat knot 6.1917

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,1,2,0,0,0,1,1,1,1,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.1917']
Arrow polynomial of the knot is: 4*K1**3 + 8*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.521', '6.920', '6.1255', '6.1917']
Outer characteristic polynomial of the knot is: t^7+28t^5+66t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1917']
2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 1024*K1**4*K2 - 1392*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 960*K1**3*K2*K3 - 192*K1**3*K3 - 256*K1**2*K2**4 + 640*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 6480*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 6968*K1**2*K2 - 336*K1**2*K3**2 - 4720*K1**2 + 2368*K1*K2**3*K3 + 544*K1*K2**2*K3*K4 - 2400*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 384*K1*K2**2*K5 - 544*K1*K2*K3*K4 + 8120*K1*K2*K3 - 32*K1*K2*K4*K5 + 1136*K1*K3*K4 + 112*K1*K4*K5 - 32*K2**6 - 128*K2**4*K3**2 - 64*K2**4*K4**2 + 256*K2**4*K4 - 2416*K2**4 + 192*K2**3*K3*K5 + 128*K2**3*K4*K6 - 128*K2**3*K6 - 2544*K2**2*K3**2 - 32*K2**2*K3*K7 - 528*K2**2*K4**2 + 2856*K2**2*K4 - 64*K2**2*K5**2 - 48*K2**2*K6**2 - 3650*K2**2 - 96*K2*K3**2*K4 + 1304*K2*K3*K5 + 280*K2*K4*K6 + 8*K2*K5*K7 - 2356*K3**2 - 820*K4**2 - 172*K5**2 - 38*K6**2 + 4202
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1917']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4188', 'vk6.4267', 'vk6.5434', 'vk6.5550', 'vk6.7549', 'vk6.7630', 'vk6.9057', 'vk6.9136', 'vk6.18248', 'vk6.18583', 'vk6.24724', 'vk6.25137', 'vk6.36854', 'vk6.37317', 'vk6.44083', 'vk6.44422', 'vk6.48500', 'vk6.48579', 'vk6.49184', 'vk6.49294', 'vk6.50285', 'vk6.50355', 'vk6.51054', 'vk6.51085', 'vk6.56043', 'vk6.56317', 'vk6.60596', 'vk6.60939', 'vk6.65713', 'vk6.66007', 'vk6.68754', 'vk6.68962']
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 O1O2O3U1U3U4O5O6O4U2U5U6
R3 orbit {'O1O2O3U1U3U4O5O6O4U2U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U5O4O6O5U2U3U6
Gauss code of K* O1O2O3U4U1U5O4O5O6U2U3U6
Gauss code of -K* O1O2O3U2U4U3O5O6O4U1U5U6
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 -1 1 2 -1 1],[ 2 0 2 1 2 1 1],[ 1 -2 0 0 1 0 1],[-1 -1 0 0 -1 0 0],[-2 -2 -1 1 0 -1 0],[ 1 -1 0 0 1 0 1],[-1 -1 -1 0 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 1 0 -1 -1 -2],[-1 -1 0 0 0 0 -1],[-1 0 0 0 -1 -1 -1],[ 1 1 0 1 0 0 -1],[ 1 1 0 1 0 0 -2],[ 2 2 1 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,-1,0,1,1,2,0,0,0,1,1,1,1,0,1,2]
Phi over symmetry [-2,-1,-1,1,1,2,-1,0,1,1,2,0,0,0,1,1,1,1,0,1,2]
Phi of -K [-2,-1,-1,1,1,2,-1,0,2,2,2,0,1,2,2,1,2,2,0,1,2]
Phi of K* [-2,-1,-1,1,1,2,1,2,2,2,2,0,1,1,2,2,2,2,0,-1,0]
Phi of -K* [-2,-1,-1,1,1,2,1,2,1,1,2,0,0,1,1,0,1,1,0,-1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+16t^4+22t^2+1
Outer characteristic polynomial t^7+28t^5+66t^3+4t
Flat arrow polynomial 4*K1**3 + 8*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -512*K1**4*K2**2 + 1024*K1**4*K2 - 1392*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 960*K1**3*K2*K3 - 192*K1**3*K3 - 256*K1**2*K2**4 + 640*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 6480*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 6968*K1**2*K2 - 336*K1**2*K3**2 - 4720*K1**2 + 2368*K1*K2**3*K3 + 544*K1*K2**2*K3*K4 - 2400*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 384*K1*K2**2*K5 - 544*K1*K2*K3*K4 + 8120*K1*K2*K3 - 32*K1*K2*K4*K5 + 1136*K1*K3*K4 + 112*K1*K4*K5 - 32*K2**6 - 128*K2**4*K3**2 - 64*K2**4*K4**2 + 256*K2**4*K4 - 2416*K2**4 + 192*K2**3*K3*K5 + 128*K2**3*K4*K6 - 128*K2**3*K6 - 2544*K2**2*K3**2 - 32*K2**2*K3*K7 - 528*K2**2*K4**2 + 2856*K2**2*K4 - 64*K2**2*K5**2 - 48*K2**2*K6**2 - 3650*K2**2 - 96*K2*K3**2*K4 + 1304*K2*K3*K5 + 280*K2*K4*K6 + 8*K2*K5*K7 - 2356*K3**2 - 820*K4**2 - 172*K5**2 - 38*K6**2 + 4202
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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