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Flat knot 6.874

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,1,1,1,2,2,0,1,1,1,0,-1,0,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.874']
Arrow polynomial of the knot is: 4*K1**3 + 8*K1**2*K2 - 8*K1**2 - 2*K1*K2 - 4*K1*K3 - 2*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.874']
Outer characteristic polynomial of the knot is: t^7+51t^5+58t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.874']
2-strand cable arrow polynomial of the knot is: -640*K1**4*K2**2 + 1216*K1**4*K2 - 1952*K1**4 - 256*K1**3*K2**2*K3 + 1440*K1**3*K2*K3 - 544*K1**3*K3 - 256*K1**2*K2**4 + 1440*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7632*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 1536*K1**2*K2*K4 + 6528*K1**2*K2 - 512*K1**2*K3**2 - 3912*K1**2 + 1952*K1*K2**3*K3 + 992*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 - 192*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7776*K1*K2*K3 - 160*K1*K2*K4*K5 + 1600*K1*K3*K4 - 32*K2**6 - 64*K2**4*K3**2 - 64*K2**4*K4**2 + 160*K2**4*K4 - 1584*K2**4 + 128*K2**3*K3*K5 + 128*K2**3*K4*K6 - 32*K2**3*K6 - 1952*K2**2*K3**2 - 936*K2**2*K4**2 + 1800*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 2680*K2**2 - 320*K2*K3**2*K4 + 816*K2*K3*K5 + 608*K2*K4*K6 + 16*K3**2*K6 - 2152*K3**2 - 864*K4**2 - 80*K5**2 - 72*K6**2 + 3614
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.874']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11603', 'vk6.11607', 'vk6.11954', 'vk6.11958', 'vk6.12949', 'vk6.12953', 'vk6.13258', 'vk6.20424', 'vk6.20426', 'vk6.21791', 'vk6.27784', 'vk6.27786', 'vk6.29306', 'vk6.31406', 'vk6.31410', 'vk6.32584', 'vk6.32588', 'vk6.32961', 'vk6.39210', 'vk6.39212', 'vk6.41434', 'vk6.47557', 'vk6.53202', 'vk6.53206', 'vk6.53511', 'vk6.57285', 'vk6.57287', 'vk6.61955', 'vk6.61957', 'vk6.64291', 'vk6.64295', 'vk6.64499']
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 O1O2O3O4U1U5O6O5U3U4U6U2
R3 orbit {'O1O2O3O4U1U5O6O5U3U4U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U1U2O6O5U6U4
Gauss code of K* O1O2O3O4U5U4U1U2O5O6U3U6
Gauss code of -K* O1O2O3O4U5U2O5O6U3U4U1U6
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 -1 1 1 1],[ 3 0 3 1 2 3 2],[-1 -3 0 -2 0 0 1],[ 1 -1 2 0 1 1 1],[-1 -2 0 -1 0 -1 0],[-1 -3 0 -1 1 0 1],[-1 -2 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 1 1 0 -1 -3],[-1 -1 0 0 0 -1 -2],[-1 -1 0 0 -1 -1 -2],[-1 0 0 1 0 -2 -3],[ 1 1 1 1 2 0 -1],[ 3 3 2 2 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,1,3,-1,-1,0,1,3,0,0,1,2,1,1,2,2,3,1]
Phi over symmetry [-3,-1,1,1,1,1,1,1,1,2,2,0,1,1,1,0,-1,0,-1,-1,0]
Phi of -K [-3,-1,1,1,1,1,1,1,1,2,2,0,1,1,1,0,-1,0,-1,-1,0]
Phi of K* [-1,-1,-1,-1,1,3,-1,-1,0,1,2,0,0,0,1,1,1,1,1,2,1]
Phi of -K* [-3,-1,1,1,1,1,1,2,2,3,3,1,1,1,2,0,-1,-1,-1,0,0]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+37t^4+34t^2
Outer characteristic polynomial t^7+51t^5+58t^3+10t
Flat arrow polynomial 4*K1**3 + 8*K1**2*K2 - 8*K1**2 - 2*K1*K2 - 4*K1*K3 - 2*K1 + 2*K2 + 3
2-strand cable arrow polynomial -640*K1**4*K2**2 + 1216*K1**4*K2 - 1952*K1**4 - 256*K1**3*K2**2*K3 + 1440*K1**3*K2*K3 - 544*K1**3*K3 - 256*K1**2*K2**4 + 1440*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7632*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 1536*K1**2*K2*K4 + 6528*K1**2*K2 - 512*K1**2*K3**2 - 3912*K1**2 + 1952*K1*K2**3*K3 + 992*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 - 192*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7776*K1*K2*K3 - 160*K1*K2*K4*K5 + 1600*K1*K3*K4 - 32*K2**6 - 64*K2**4*K3**2 - 64*K2**4*K4**2 + 160*K2**4*K4 - 1584*K2**4 + 128*K2**3*K3*K5 + 128*K2**3*K4*K6 - 32*K2**3*K6 - 1952*K2**2*K3**2 - 936*K2**2*K4**2 + 1800*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 2680*K2**2 - 320*K2*K3**2*K4 + 816*K2*K3*K5 + 608*K2*K4*K6 + 16*K3**2*K6 - 2152*K3**2 - 864*K4**2 - 80*K5**2 - 72*K6**2 + 3614
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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