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Flat knot 6.389

Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,-1,-1,2,2,3,0,2,1,2,1,1,1,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.389', '7.33541']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.315', '6.337', '6.389', '6.418', '6.599', '6.675', '6.686', '6.688', '6.746', '6.747', '6.809', '6.1034', '6.1128', '6.1133', '6.1334', '6.1363', '6.1489', '6.1539', '6.1564', '6.1821', '6.1863']
Outer characteristic polynomial of the knot is: t^7+51t^5+34t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.389']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 512*K1**4*K2 - 928*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 + 448*K1**2*K2**3 - 3792*K1**2*K2**2 - 576*K1**2*K2*K4 + 5016*K1**2*K2 - 160*K1**2*K3**2 - 144*K1**2*K4**2 - 3776*K1**2 + 224*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 192*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 3648*K1*K2*K3 + 832*K1*K3*K4 + 136*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 592*K2**4 - 272*K2**2*K3**2 - 176*K2**2*K4**2 + 912*K2**2*K4 - 2518*K2**2 + 104*K2*K3*K5 + 48*K2*K4*K6 - 1056*K3**2 - 564*K4**2 - 56*K5**2 - 2*K6**2 + 2858
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.389']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11284', 'vk6.11362', 'vk6.12545', 'vk6.12656', 'vk6.18367', 'vk6.18706', 'vk6.24813', 'vk6.25272', 'vk6.30966', 'vk6.31091', 'vk6.32144', 'vk6.32263', 'vk6.36995', 'vk6.37446', 'vk6.44174', 'vk6.44494', 'vk6.52036', 'vk6.52123', 'vk6.52879', 'vk6.52944', 'vk6.56139', 'vk6.56366', 'vk6.60658', 'vk6.61005', 'vk6.63657', 'vk6.63702', 'vk6.64085', 'vk6.64130', 'vk6.65793', 'vk6.66050', 'vk6.68792', 'vk6.69001']
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 O1O2O3O4O5U4U2O6U5U1U6U3
R3 orbit {'O1O2O3O4O5U4U2O6U5U1U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U6U5U1O6U4U2
Gauss code of K* O1O2O3O4U2U5U4U6U1O6O5U3
Gauss code of -K* O1O2O3O4U2O5O6U4U6U1U5U3
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 -2 -2 2 -1 1 2],[ 2 0 -1 3 -1 2 2],[ 2 1 0 2 0 2 1],[-2 -3 -2 0 -1 0 1],[ 1 1 0 1 0 1 1],[-1 -2 -2 0 -1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 1 -1 -2 -2],[-2 0 1 0 -1 -2 -3],[-2 -1 0 -1 -1 -1 -2],[-1 0 1 0 -1 -2 -2],[ 1 1 1 1 0 0 1],[ 2 2 1 2 0 0 1],[ 2 3 2 2 -1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,2,2,-1,0,1,2,3,1,1,1,2,1,2,2,0,-1,-1]
Phi over symmetry [-2,-2,-1,1,2,2,-1,-1,2,2,3,0,2,1,2,1,1,1,1,0,-1]
Phi of -K [-2,-2,-1,1,2,2,-1,1,1,2,3,2,1,1,2,1,2,2,1,0,-1]
Phi of K* [-2,-2,-1,1,2,2,-1,0,2,2,3,1,2,1,2,1,1,1,2,1,-1]
Phi of -K* [-2,-2,-1,1,2,2,-1,-1,2,2,3,0,2,1,2,1,1,1,1,0,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2+21w^2z+27w
Inner characteristic polynomial t^6+33t^4+16t^2+1
Outer characteristic polynomial t^7+51t^5+34t^3+5t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -192*K1**4*K2**2 + 512*K1**4*K2 - 928*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 + 448*K1**2*K2**3 - 3792*K1**2*K2**2 - 576*K1**2*K2*K4 + 5016*K1**2*K2 - 160*K1**2*K3**2 - 144*K1**2*K4**2 - 3776*K1**2 + 224*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 192*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 3648*K1*K2*K3 + 832*K1*K3*K4 + 136*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 592*K2**4 - 272*K2**2*K3**2 - 176*K2**2*K4**2 + 912*K2**2*K4 - 2518*K2**2 + 104*K2*K3*K5 + 48*K2*K4*K6 - 1056*K3**2 - 564*K4**2 - 56*K5**2 - 2*K6**2 + 2858
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}]]
If K is slice True
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