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Flat knot 6.371

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,1,2,1,3,0,1,1,1,1,2,1,2,1,-2]
Flat knots (up to 7 crossings) with same phi are :['6.371']
Arrow polynomial of the knot is: 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.122', '6.327', '6.371', '6.1185']
Outer characteristic polynomial of the knot is: t^7+51t^5+99t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.371']
2-strand cable arrow polynomial of the knot is: 384*K1**4*K2**3 - 960*K1**4*K2**2 + 1024*K1**4*K2 - 1440*K1**4 + 160*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1024*K1**2*K2**4 + 1632*K1**2*K2**3 - 4400*K1**2*K2**2 + 3784*K1**2*K2 - 160*K1**2*K3**2 - 64*K1**2*K4**2 - 2648*K1**2 + 1536*K1*K2**3*K3 + 3928*K1*K2*K3 + 496*K1*K3*K4 + 64*K1*K4*K5 + 8*K1*K5*K6 - 192*K2**6 - 448*K2**4*K3**2 - 192*K2**4*K4**2 + 672*K2**4*K4 - 2096*K2**4 + 352*K2**3*K3*K5 + 128*K2**3*K4*K6 - 1376*K2**2*K3**2 - 536*K2**2*K4**2 + 1072*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 1182*K2**2 + 616*K2*K3*K5 + 192*K2*K4*K6 + 8*K2*K5*K7 + 8*K3**2*K6 - 1320*K3**2 - 552*K4**2 - 112*K5**2 - 34*K6**2 + 2734
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.371']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4749', 'vk6.5078', 'vk6.6291', 'vk6.6732', 'vk6.8252', 'vk6.8703', 'vk6.9638', 'vk6.9955', 'vk6.20403', 'vk6.21756', 'vk6.27743', 'vk6.29277', 'vk6.39177', 'vk6.41409', 'vk6.45907', 'vk6.47544', 'vk6.48781', 'vk6.48994', 'vk6.49593', 'vk6.49798', 'vk6.50793', 'vk6.51010', 'vk6.51280', 'vk6.51477', 'vk6.57272', 'vk6.58501', 'vk6.61926', 'vk6.63023', 'vk6.66885', 'vk6.67767', 'vk6.69519', 'vk6.70229']
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 O1O2O3O4O5U3U5O6U4U1U2U6
R3 orbit {'O1O2O3O4O5U3U5O6U4U1U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U4U5U2O6U1U3
Gauss code of K* O1O2O3O4U2U3U5U1U6O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U5U4U6U2U3
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 0 -2 0 1 3],[ 2 0 1 -2 1 1 3],[ 0 -1 0 -2 1 1 2],[ 2 2 2 0 2 1 1],[ 0 -1 -1 -2 0 0 1],[-1 -1 -1 -1 0 0 0],[-3 -3 -2 -1 -1 0 0]]
Primitive based matrix [[ 0 3 1 0 0 -2 -2],[-3 0 0 -1 -2 -1 -3],[-1 0 0 0 -1 -1 -1],[ 0 1 0 0 -1 -2 -1],[ 0 2 1 1 0 -2 -1],[ 2 1 1 2 2 0 2],[ 2 3 1 1 1 -2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,2,2,0,1,2,1,3,0,1,1,1,1,2,1,2,1,-2]
Phi over symmetry [-3,-1,0,0,2,2,0,1,2,1,3,0,1,1,1,1,2,1,2,1,-2]
Phi of -K [-2,-2,0,0,1,3,-2,0,0,2,4,1,1,2,2,-1,0,1,1,2,2]
Phi of K* [-3,-1,0,0,2,2,2,1,2,2,4,0,1,2,2,1,1,0,1,0,-2]
Phi of -K* [-2,-2,0,0,1,3,-2,1,1,1,3,2,2,1,1,-1,0,1,1,2,0]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 11z+23
Enhanced Jones-Krushkal polynomial -8w^3z+19w^2z+23w
Inner characteristic polynomial t^6+33t^4+31t^2
Outer characteristic polynomial t^7+51t^5+99t^3
Flat arrow polynomial 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial 384*K1**4*K2**3 - 960*K1**4*K2**2 + 1024*K1**4*K2 - 1440*K1**4 + 160*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1024*K1**2*K2**4 + 1632*K1**2*K2**3 - 4400*K1**2*K2**2 + 3784*K1**2*K2 - 160*K1**2*K3**2 - 64*K1**2*K4**2 - 2648*K1**2 + 1536*K1*K2**3*K3 + 3928*K1*K2*K3 + 496*K1*K3*K4 + 64*K1*K4*K5 + 8*K1*K5*K6 - 192*K2**6 - 448*K2**4*K3**2 - 192*K2**4*K4**2 + 672*K2**4*K4 - 2096*K2**4 + 352*K2**3*K3*K5 + 128*K2**3*K4*K6 - 1376*K2**2*K3**2 - 536*K2**2*K4**2 + 1072*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 1182*K2**2 + 616*K2*K3*K5 + 192*K2*K4*K6 + 8*K2*K5*K7 + 8*K3**2*K6 - 1320*K3**2 - 552*K4**2 - 112*K5**2 - 34*K6**2 + 2734
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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