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Flat knot 6.929

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,1,2,2,1,1,1,1,-1,0,0,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.929']
Arrow polynomial of the knot is: 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.218', '6.554', '6.929', '6.932', '6.1014', '6.1024', '6.1068', '6.1526', '6.1664', '6.1676', '6.1755', '6.1763', '6.2065', '6.2078']
Outer characteristic polynomial of the knot is: t^7+31t^5+63t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.929']
2-strand cable arrow polynomial of the knot is: -768*K1**4*K2**2 + 1504*K1**4*K2 - 3232*K1**4 + 448*K1**3*K2*K3 - 448*K1**3*K3 - 576*K1**2*K2**4 + 2080*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7824*K1**2*K2**2 - 512*K1**2*K2*K4 + 8336*K1**2*K2 - 352*K1**2*K3**2 - 3516*K1**2 + 1120*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 224*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 6176*K1*K2*K3 + 608*K1*K3*K4 + 56*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 2048*K2**4 - 64*K2**3*K6 - 672*K2**2*K3**2 - 128*K2**2*K4**2 + 1544*K2**2*K4 - 2380*K2**2 + 384*K2*K3*K5 + 80*K2*K4*K6 - 1264*K3**2 - 344*K4**2 - 76*K5**2 - 12*K6**2 + 3222
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.929']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13371', 'vk6.13440', 'vk6.13629', 'vk6.13751', 'vk6.14152', 'vk6.14381', 'vk6.15612', 'vk6.16078', 'vk6.16468', 'vk6.16483', 'vk6.17642', 'vk6.22875', 'vk6.22906', 'vk6.24195', 'vk6.33126', 'vk6.33165', 'vk6.33227', 'vk6.33284', 'vk6.34856', 'vk6.34887', 'vk6.36446', 'vk6.42442', 'vk6.42457', 'vk6.43548', 'vk6.53547', 'vk6.53580', 'vk6.53611', 'vk6.53679', 'vk6.54722', 'vk6.55680', 'vk6.60234', 'vk6.64573']
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 O1O2O3O4U3U5O6O5U4U1U2U6
R3 orbit {'O1O2O3O4U3U5O6O5U4U1U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3U4U1O6O5U6U2
Gauss code of K* O1O2O3O4U2U3U5U1O5O6U4U6
Gauss code of -K* O1O2O3O4U5U1O5O6U4U6U2U3
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 0 -1 0 1 2],[ 2 0 1 -1 1 2 2],[ 0 -1 0 -1 1 0 1],[ 1 1 1 0 1 1 1],[ 0 -1 -1 -1 0 0 0],[-1 -2 0 -1 0 0 2],[-2 -2 -1 -1 0 -2 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -2 0 -1 -1 -2],[-1 2 0 0 0 -1 -2],[ 0 0 0 0 -1 -1 -1],[ 0 1 0 1 0 -1 -1],[ 1 1 1 1 1 0 1],[ 2 2 2 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,2,0,1,1,2,0,0,1,2,1,1,1,1,1,-1]
Phi over symmetry [-2,-1,0,0,1,2,-1,1,1,2,2,1,1,1,1,-1,0,0,0,1,2]
Phi of -K [-2,-1,0,0,1,2,2,1,1,1,2,0,0,1,2,-1,1,1,1,2,-1]
Phi of K* [-2,-1,0,0,1,2,-1,1,2,2,2,1,1,1,1,1,0,1,0,1,2]
Phi of -K* [-2,-1,0,0,1,2,-1,1,1,2,2,1,1,1,1,-1,0,0,0,1,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 3z^2+20z+29
Enhanced Jones-Krushkal polynomial 3w^3z^2+20w^2z+29w
Inner characteristic polynomial t^6+21t^4+27t^2+1
Outer characteristic polynomial t^7+31t^5+63t^3+5t
Flat arrow polynomial 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -768*K1**4*K2**2 + 1504*K1**4*K2 - 3232*K1**4 + 448*K1**3*K2*K3 - 448*K1**3*K3 - 576*K1**2*K2**4 + 2080*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7824*K1**2*K2**2 - 512*K1**2*K2*K4 + 8336*K1**2*K2 - 352*K1**2*K3**2 - 3516*K1**2 + 1120*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 224*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 6176*K1*K2*K3 + 608*K1*K3*K4 + 56*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 2048*K2**4 - 64*K2**3*K6 - 672*K2**2*K3**2 - 128*K2**2*K4**2 + 1544*K2**2*K4 - 2380*K2**2 + 384*K2*K3*K5 + 80*K2*K4*K6 - 1264*K3**2 - 344*K4**2 - 76*K5**2 - 12*K6**2 + 3222
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {2}]]
If K is slice False
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