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Flat knot 6.976

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,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.976']
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+52t^5+33t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.976']
2-strand cable arrow polynomial of the knot is: -192*K1**6 - 192*K1**4*K2**2 + 1568*K1**4*K2 - 6112*K1**4 + 832*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1248*K1**3*K3 - 5856*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 12792*K1**2*K2 - 1280*K1**2*K3**2 - 112*K1**2*K4**2 - 6868*K1**2 - 480*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 8792*K1*K2*K3 + 1440*K1*K3*K4 + 72*K1*K4*K5 - 344*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 792*K2**2*K4 - 6190*K2**2 + 128*K2*K3*K5 + 8*K2*K4*K6 - 2880*K3**2 - 606*K4**2 - 52*K5**2 - 2*K6**2 + 6348
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.976']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20022', 'vk6.20064', 'vk6.21294', 'vk6.21346', 'vk6.27073', 'vk6.27129', 'vk6.28778', 'vk6.28818', 'vk6.38470', 'vk6.38522', 'vk6.40659', 'vk6.40719', 'vk6.45354', 'vk6.45422', 'vk6.47123', 'vk6.47164', 'vk6.56821', 'vk6.56885', 'vk6.57955', 'vk6.58023', 'vk6.61339', 'vk6.61415', 'vk6.62515', 'vk6.62572', 'vk6.66541', 'vk6.66585', 'vk6.67330', 'vk6.67376', 'vk6.69187', 'vk6.69237', 'vk6.69938', 'vk6.69978']
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 O1O2O3O4U1U3O5U2O6U4U6U5
R3 orbit {'O1O2O3O4U1U3O5U2O6U4U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U1O6U3O5U2U4
Gauss code of K* O1O2O3U4U5U6U1O4O6U3O5U2
Gauss code of -K* O1O2O3U2O4U1O5O6U3U5U4U6
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 2 1],[ 3 0 2 1 3 2 1],[ 1 -2 0 0 2 2 1],[ 0 -1 0 0 1 1 1],[-1 -3 -2 -1 0 2 1],[-2 -2 -2 -1 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -2 -1 -2 -2],[-1 0 0 -1 -1 -1 -1],[-1 2 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,-1,0,1,3,0,2,1,2,2,1,1,1,1,1,2,3,0,1,2]
Phi over symmetry [-3,-1,0,1,1,2,0,2,1,3,3,1,0,1,1,0,0,1,-1,-1,1]
Phi of -K [-3,-1,0,1,1,2,0,2,1,3,3,1,0,1,1,0,0,1,-1,-1,1]
Phi of K* [-2,-1,-1,0,1,3,-1,1,1,1,3,1,0,0,1,0,1,3,1,2,0]
Phi of -K* [-3,-1,0,1,1,2,2,1,1,3,2,0,1,2,2,1,1,1,-1,0,2]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial z^2+22z+41
Enhanced Jones-Krushkal polynomial w^3z^2+22w^2z+41w
Inner characteristic polynomial t^6+36t^4+12t^2+1
Outer characteristic polynomial t^7+52t^5+33t^3+4t
Flat arrow polynomial -14*K1**2 - 2*K1*K2 + K1 + 7*K2 + K3 + 8
2-strand cable arrow polynomial -192*K1**6 - 192*K1**4*K2**2 + 1568*K1**4*K2 - 6112*K1**4 + 832*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1248*K1**3*K3 - 5856*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 12792*K1**2*K2 - 1280*K1**2*K3**2 - 112*K1**2*K4**2 - 6868*K1**2 - 480*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 8792*K1*K2*K3 + 1440*K1*K3*K4 + 72*K1*K4*K5 - 344*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 792*K2**2*K4 - 6190*K2**2 + 128*K2*K3*K5 + 8*K2*K4*K6 - 2880*K3**2 - 606*K4**2 - 52*K5**2 - 2*K6**2 + 6348
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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