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Flat knot 6.1228

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,-1,1,2,2,0,0,0,1,0,0,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1228']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1*K3 + K4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.111', '6.139', '6.519', '6.566', '6.1228', '6.1254', '6.1259', '6.1912', '6.1936']
Outer characteristic polynomial of the knot is: t^7+28t^5+64t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1204', '6.1228']
2-strand cable arrow polynomial of the knot is: 768*K1**4*K2 - 1536*K1**4 - 384*K1**3*K2**2*K3 + 768*K1**3*K2*K3 - 320*K1**3*K3 + 1344*K1**2*K2**3 - 4352*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 992*K1**2*K2*K4 + 4048*K1**2*K2 - 1664*K1**2*K3**2 - 320*K1**2*K3*K5 - 192*K1**2*K4**2 - 2496*K1**2 + 544*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1728*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 - 544*K1*K2*K3*K4 - 224*K1*K2*K3*K6 + 5592*K1*K2*K3 - 64*K1*K2*K4*K7 - 32*K1*K2*K5*K6 + 2288*K1*K3*K4 + 544*K1*K4*K5 + 48*K1*K5*K6 + 24*K1*K6*K7 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1200*K2**4 + 32*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1120*K2**2*K3**2 - 288*K2**2*K4**2 + 1600*K2**2*K4 - 96*K2**2*K5**2 - 48*K2**2*K6**2 - 2008*K2**2 + 960*K2*K3*K5 + 256*K2*K4*K6 + 72*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1512*K3**2 - 824*K4**2 - 280*K5**2 - 56*K6**2 - 16*K7**2 - 2*K8**2 + 2456
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1228']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4884', 'vk6.4891', 'vk6.5227', 'vk6.5236', 'vk6.6479', 'vk6.6490', 'vk6.6897', 'vk6.8446', 'vk6.8863', 'vk6.8870', 'vk6.9783', 'vk6.9786', 'vk6.10074', 'vk6.10079', 'vk6.20821', 'vk6.20831', 'vk6.22217', 'vk6.29782', 'vk6.29792', 'vk6.39881', 'vk6.39887', 'vk6.46436', 'vk6.47988', 'vk6.47994', 'vk6.48842', 'vk6.49111', 'vk6.49116', 'vk6.51365', 'vk6.51368', 'vk6.51576', 'vk6.63280', 'vk6.67121']
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 O1O2O3O4U3U2U4O5O6U1U6U5
R3 orbit {'O1O2O3O4U3U2U4O5O6U1U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U4O6O5U1U3U2
Gauss code of K* O1O2O3U4U5O6O5O4U6U2U1U3
Gauss code of -K* O1O2O3U2U1O4O5O6U4U6U5U3
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 -1 -1 2 2 1],[ 1 1 0 0 2 0 0],[ 1 1 0 0 1 0 0],[-2 -2 -2 -1 0 0 0],[-1 -2 0 0 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 0 -1 -2 -2],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 0 -2],[ 1 1 0 0 0 0 1],[ 1 2 0 0 0 0 1],[ 2 2 1 2 -1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,0,1,2,2,0,0,0,1,0,0,2,0,-1,-1]
Phi over symmetry [-2,-1,-1,1,1,2,-1,-1,1,2,2,0,0,0,1,0,0,2,0,0,0]
Phi of -K [-2,-1,-1,1,1,2,2,2,1,2,2,0,2,2,1,2,2,2,0,1,1]
Phi of K* [-2,-1,-1,1,1,2,1,1,1,2,2,0,2,2,1,2,2,2,0,2,2]
Phi of -K* [-2,-1,-1,1,1,2,-1,-1,1,2,2,0,0,0,1,0,0,2,0,0,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+16t^4+26t^2
Outer characteristic polynomial t^7+28t^5+64t^3+6t
Flat arrow polynomial 4*K1**2*K2 - 4*K1*K3 + K4
2-strand cable arrow polynomial 768*K1**4*K2 - 1536*K1**4 - 384*K1**3*K2**2*K3 + 768*K1**3*K2*K3 - 320*K1**3*K3 + 1344*K1**2*K2**3 - 4352*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 992*K1**2*K2*K4 + 4048*K1**2*K2 - 1664*K1**2*K3**2 - 320*K1**2*K3*K5 - 192*K1**2*K4**2 - 2496*K1**2 + 544*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1728*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 - 544*K1*K2*K3*K4 - 224*K1*K2*K3*K6 + 5592*K1*K2*K3 - 64*K1*K2*K4*K7 - 32*K1*K2*K5*K6 + 2288*K1*K3*K4 + 544*K1*K4*K5 + 48*K1*K5*K6 + 24*K1*K6*K7 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1200*K2**4 + 32*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1120*K2**2*K3**2 - 288*K2**2*K4**2 + 1600*K2**2*K4 - 96*K2**2*K5**2 - 48*K2**2*K6**2 - 2008*K2**2 + 960*K2*K3*K5 + 256*K2*K4*K6 + 72*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1512*K3**2 - 824*K4**2 - 280*K5**2 - 56*K6**2 - 16*K7**2 - 2*K8**2 + 2456
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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