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Flat knot 6.865

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,3,1,0,0,1,1,0,1,2,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.865']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 6*K1*K2 - 4*K1*K3 + 3*K2 + 2*K3 + 2*K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.865']
Outer characteristic polynomial of the knot is: t^7+42t^5+66t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.865']
2-strand cable arrow polynomial of the knot is: 1504*K1**4*K2 - 5200*K1**4 + 1536*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1184*K1**3*K3 - 128*K1**2*K2**4 + 544*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 6176*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 1248*K1**2*K2*K4 + 9608*K1**2*K2 - 2768*K1**2*K3**2 - 288*K1**2*K3*K5 - 192*K1**2*K4**2 - 64*K1**2*K5**2 - 5000*K1**2 + 448*K1*K2**3*K3 - 1696*K1*K2**2*K3 - 544*K1*K2**2*K5 + 32*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 224*K1*K2*K3*K6 + 9848*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 3672*K1*K3*K4 + 696*K1*K4*K5 + 248*K1*K5*K6 + 32*K1*K6*K7 - 32*K2**6 + 64*K2**4*K4 - 808*K2**4 - 96*K2**3*K6 - 800*K2**2*K3**2 - 32*K2**2*K3*K7 - 24*K2**2*K4**2 + 2040*K2**2*K4 - 16*K2**2*K5**2 - 16*K2**2*K6**2 - 5468*K2**2 + 1448*K2*K3*K5 + 272*K2*K4*K6 + 88*K2*K5*K7 + 32*K2*K6*K8 - 96*K3**4 + 200*K3**2*K6 - 3476*K3**2 + 24*K3*K4*K7 - 1482*K4**2 - 648*K5**2 - 252*K6**2 - 52*K7**2 - 12*K8**2 + 5780
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.865']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4355', 'vk6.4388', 'vk6.5677', 'vk6.5710', 'vk6.7738', 'vk6.7771', 'vk6.9220', 'vk6.9253', 'vk6.10485', 'vk6.10567', 'vk6.10662', 'vk6.10706', 'vk6.10739', 'vk6.10849', 'vk6.14614', 'vk6.15313', 'vk6.15438', 'vk6.16233', 'vk6.17978', 'vk6.24418', 'vk6.30164', 'vk6.30246', 'vk6.30341', 'vk6.30468', 'vk6.33959', 'vk6.34360', 'vk6.34414', 'vk6.43845', 'vk6.50425', 'vk6.50457', 'vk6.54218', 'vk6.63424']
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 O1O2O3O4U1U4O5O6U5U3U6U2
R3 orbit {'O1O2O3O4U1U4O5O6U5U3U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U2U6O5O6U1U4
Gauss code of K* O1O2O3O4U5U4U2U6O5O6U1U3
Gauss code of -K* O1O2O3O4U2U4O5O6U5U3U1U6
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 -3 1 0 1 -1 2],[ 3 0 3 2 1 0 1],[-1 -3 0 -1 0 -1 2],[ 0 -2 1 0 0 0 2],[-1 -1 0 0 0 0 0],[ 1 0 1 0 0 0 1],[-2 -1 -2 -2 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -2 -2 -1 -1],[-1 0 0 0 0 0 -1],[-1 2 0 0 -1 -1 -3],[ 0 2 0 1 0 0 -2],[ 1 1 0 1 0 0 0],[ 3 1 1 3 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,2,2,1,1,0,0,0,1,1,1,3,0,2,0]
Phi over symmetry [-3,-1,0,1,1,2,0,2,1,3,1,0,0,1,1,0,1,2,0,0,2]
Phi of -K [-3,-1,0,1,1,2,2,1,1,3,4,1,1,2,2,0,1,0,0,-1,1]
Phi of K* [-2,-1,-1,0,1,3,-1,1,0,2,4,0,0,1,1,1,2,3,1,1,2]
Phi of -K* [-3,-1,0,1,1,2,0,2,1,3,1,0,0,1,1,0,1,2,0,0,2]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+26t^4+25t^2+1
Outer characteristic polynomial t^7+42t^5+66t^3+5t
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 6*K1*K2 - 4*K1*K3 + 3*K2 + 2*K3 + 2*K4 + 2
2-strand cable arrow polynomial 1504*K1**4*K2 - 5200*K1**4 + 1536*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1184*K1**3*K3 - 128*K1**2*K2**4 + 544*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 6176*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 1248*K1**2*K2*K4 + 9608*K1**2*K2 - 2768*K1**2*K3**2 - 288*K1**2*K3*K5 - 192*K1**2*K4**2 - 64*K1**2*K5**2 - 5000*K1**2 + 448*K1*K2**3*K3 - 1696*K1*K2**2*K3 - 544*K1*K2**2*K5 + 32*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 224*K1*K2*K3*K6 + 9848*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 3672*K1*K3*K4 + 696*K1*K4*K5 + 248*K1*K5*K6 + 32*K1*K6*K7 - 32*K2**6 + 64*K2**4*K4 - 808*K2**4 - 96*K2**3*K6 - 800*K2**2*K3**2 - 32*K2**2*K3*K7 - 24*K2**2*K4**2 + 2040*K2**2*K4 - 16*K2**2*K5**2 - 16*K2**2*K6**2 - 5468*K2**2 + 1448*K2*K3*K5 + 272*K2*K4*K6 + 88*K2*K5*K7 + 32*K2*K6*K8 - 96*K3**4 + 200*K3**2*K6 - 3476*K3**2 + 24*K3*K4*K7 - 1482*K4**2 - 648*K5**2 - 252*K6**2 - 52*K7**2 - 12*K8**2 + 5780
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}]]
If K is slice False
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