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Flat knot 6.815

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,3,3,1,0,1,1,0,0,1,-1,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.815']
Arrow polynomial of the knot is: -14*K1**2 - 2*K1*K2 + K1 + 7*K2 + K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.584', '6.815', '6.976']
Outer characteristic polynomial of the knot is: t^7+50t^5+44t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.815']
2-strand cable arrow polynomial of the knot is: -832*K1**6 - 576*K1**4*K2**2 + 2400*K1**4*K2 - 7248*K1**4 + 640*K1**3*K2*K3 - 1056*K1**3*K3 - 6624*K1**2*K2**2 - 416*K1**2*K2*K4 + 13320*K1**2*K2 - 528*K1**2*K3**2 - 32*K1**2*K3*K5 - 64*K1**2*K4**2 - 5164*K1**2 - 608*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 7744*K1*K2*K3 + 1264*K1*K3*K4 + 136*K1*K4*K5 - 344*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 712*K2**2*K4 - 5246*K2**2 + 88*K2*K3*K5 + 8*K2*K4*K6 - 2284*K3**2 - 570*K4**2 - 64*K5**2 - 2*K6**2 + 5448
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.815']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4126', 'vk6.4157', 'vk6.5364', 'vk6.5395', 'vk6.7486', 'vk6.7517', 'vk6.8987', 'vk6.9018', 'vk6.12446', 'vk6.12479', 'vk6.13340', 'vk6.13561', 'vk6.13592', 'vk6.14251', 'vk6.14698', 'vk6.14755', 'vk6.15210', 'vk6.15854', 'vk6.15909', 'vk6.30851', 'vk6.30884', 'vk6.32035', 'vk6.32068', 'vk6.33058', 'vk6.33089', 'vk6.33861', 'vk6.34320', 'vk6.48486', 'vk6.50265', 'vk6.53528', 'vk6.53942', 'vk6.54270']
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 O1O2O3O4U1O5U4U5U6U3O6U2
R3 orbit {'O1O2O3O4U1O5U4U5U6U3O6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3O5U2U5U6U1O6U4
Gauss code of K* O1O2O3O4U3O5U6U5U4U1O6U2
Gauss code of -K* O1O2O3O4U3O5U4U1U6U5O6U2
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 2 0 1 -1],[ 3 0 3 2 1 1 2],[-1 -3 0 1 -1 1 -2],[-2 -2 -1 0 -1 1 -2],[ 0 -1 1 1 0 1 0],[-1 -1 -1 -1 -1 0 -1],[ 1 -2 2 2 0 1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 1 -1 -1 -2 -2],[-1 -1 0 -1 -1 -1 -1],[-1 1 1 0 -1 -2 -3],[ 0 1 1 1 0 0 -1],[ 1 2 1 2 0 0 -2],[ 3 2 1 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,-1,1,1,2,2,1,1,1,1,1,2,3,0,1,2]
Phi over symmetry [-3,-1,0,1,1,2,0,2,1,3,3,1,0,1,1,0,0,1,-1,0,2]
Phi of -K [-3,-1,0,1,1,2,0,2,1,3,3,1,0,1,1,0,0,1,-1,0,2]
Phi of K* [-2,-1,-1,0,1,3,0,2,1,1,3,1,0,0,1,0,1,3,1,2,0]
Phi of -K* [-3,-1,0,1,1,2,2,1,1,3,2,0,1,2,2,1,1,1,-1,-1,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+34t^4+15t^2
Outer characteristic polynomial t^7+50t^5+44t^3+4t
Flat arrow polynomial -14*K1**2 - 2*K1*K2 + K1 + 7*K2 + K3 + 8
2-strand cable arrow polynomial -832*K1**6 - 576*K1**4*K2**2 + 2400*K1**4*K2 - 7248*K1**4 + 640*K1**3*K2*K3 - 1056*K1**3*K3 - 6624*K1**2*K2**2 - 416*K1**2*K2*K4 + 13320*K1**2*K2 - 528*K1**2*K3**2 - 32*K1**2*K3*K5 - 64*K1**2*K4**2 - 5164*K1**2 - 608*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 7744*K1*K2*K3 + 1264*K1*K3*K4 + 136*K1*K4*K5 - 344*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 712*K2**2*K4 - 5246*K2**2 + 88*K2*K3*K5 + 8*K2*K4*K6 - 2284*K3**2 - 570*K4**2 - 64*K5**2 - 2*K6**2 + 5448
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {1, 5}, {2, 4}]]
If K is slice False
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