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Flat knot 6.1256

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,1,2,0,1,1,2,0,0,1,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.1256']
Arrow polynomial of the knot is: 4*K1**2*K2 - 8*K1**2 - 4*K1*K3 + 4*K2 + K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1256']
Outer characteristic polynomial of the knot is: t^7+29t^5+29t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1256']
2-strand cable arrow polynomial of the knot is: -1952*K1**4 - 448*K1**3*K3 - 2080*K1**2*K2**2 - 704*K1**2*K2*K4 + 5632*K1**2*K2 - 1088*K1**2*K3**2 - 4584*K1**2 + 256*K1*K2**2*K3*K4 - 256*K1*K2**2*K3 + 64*K1*K2*K3**3 - 320*K1*K2*K3*K4 - 320*K1*K2*K3*K6 + 6080*K1*K2*K3 + 2048*K1*K3*K4 + 48*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 64*K2**4*K4 - 272*K2**4 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 512*K2**2*K3**2 - 288*K2**2*K4**2 + 1152*K2**2*K4 - 48*K2**2*K6**2 - 3984*K2**2 + 608*K2*K3*K5 + 288*K2*K4*K6 + 16*K2*K6*K8 - 32*K3**4 + 64*K3**2*K6 - 2440*K3**2 - 944*K4**2 - 96*K5**2 - 64*K6**2 - 2*K8**2 + 4112
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1256']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72428', 'vk6.72434', 'vk6.72478', 'vk6.72490', 'vk6.72843', 'vk6.72855', 'vk6.72902', 'vk6.74453', 'vk6.74463', 'vk6.75068', 'vk6.76966', 'vk6.77781', 'vk6.77973', 'vk6.79453', 'vk6.79462', 'vk6.79903', 'vk6.79913', 'vk6.80927', 'vk6.87234', 'vk6.89361']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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 O1O2O3O4U5U3U1O5O6U4U2U6
R3 orbit {'O1O2O3O4U5U3U1O5O6U4U2U6'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2O3U1U4O5O6O4U3U6U2U5
Gauss code of -K* O1O2O3U1U4O5O6O4U3U6U2U5
Diagrammatic symmetry type r
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 0 0 1 -2 2],[ 1 0 1 0 1 0 2],[ 0 -1 0 0 1 -1 2],[ 0 0 0 0 0 0 1],[-1 -1 -1 0 0 -1 1],[ 2 0 1 0 1 0 2],[-2 -2 -2 -1 -1 -2 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -2 -2 -2],[-1 1 0 0 -1 -1 -1],[ 0 1 0 0 0 0 0],[ 0 2 1 0 0 -1 -1],[ 1 2 1 0 1 0 0],[ 2 2 1 0 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,1,2,2,2,0,1,1,1,0,0,0,1,1,0]
Phi over symmetry [-2,-1,0,0,1,2,0,0,1,1,2,0,1,1,2,0,0,1,1,2,1]
Phi of -K [-2,-1,0,0,1,2,1,1,2,2,2,0,1,1,1,0,0,0,1,1,0]
Phi of K* [-2,-1,0,0,1,2,0,0,1,1,2,0,1,1,2,0,0,1,1,2,1]
Phi of -K* [-2,-1,0,0,1,2,0,0,1,1,2,0,1,1,2,0,0,1,1,2,1]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial z^2+18z+33
Enhanced Jones-Krushkal polynomial w^3z^2+18w^2z+33w
Inner characteristic polynomial t^6+19t^4+7t^2
Outer characteristic polynomial t^7+29t^5+29t^3+4t
Flat arrow polynomial 4*K1**2*K2 - 8*K1**2 - 4*K1*K3 + 4*K2 + K4 + 4
2-strand cable arrow polynomial -1952*K1**4 - 448*K1**3*K3 - 2080*K1**2*K2**2 - 704*K1**2*K2*K4 + 5632*K1**2*K2 - 1088*K1**2*K3**2 - 4584*K1**2 + 256*K1*K2**2*K3*K4 - 256*K1*K2**2*K3 + 64*K1*K2*K3**3 - 320*K1*K2*K3*K4 - 320*K1*K2*K3*K6 + 6080*K1*K2*K3 + 2048*K1*K3*K4 + 48*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 64*K2**4*K4 - 272*K2**4 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 512*K2**2*K3**2 - 288*K2**2*K4**2 + 1152*K2**2*K4 - 48*K2**2*K6**2 - 3984*K2**2 + 608*K2*K3*K5 + 288*K2*K4*K6 + 16*K2*K6*K8 - 32*K3**4 + 64*K3**2*K6 - 2440*K3**2 - 944*K4**2 - 96*K5**2 - 64*K6**2 - 2*K8**2 + 4112
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]]
If K is slice False
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