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Flat knot 6.208

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,0,2,2,4,3,1,1,1,1,1,2,2,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.208']
Arrow polynomial of the knot is: -6*K1**2 - 6*K1*K2 + 3*K1 - 2*K2**2 + 3*K2 + 3*K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.208']
Outer characteristic polynomial of the knot is: t^7+80t^5+44t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.208']
2-strand cable arrow polynomial of the knot is: 832*K1**4*K2 - 2384*K1**4 + 864*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1504*K1**3*K3 + 288*K1**2*K2**2*K4 - 3600*K1**2*K2**2 - 832*K1**2*K2*K4 + 7720*K1**2*K2 - 2224*K1**2*K3**2 - 256*K1**2*K4**2 - 6520*K1**2 + 192*K1*K2**3*K3 - 864*K1*K2**2*K3 - 384*K1*K2**2*K5 + 224*K1*K2*K3**3 + 96*K1*K2*K3*K4**2 - 832*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 8984*K1*K2*K3 + 3320*K1*K3*K4 + 728*K1*K4*K5 - 216*K2**4 - 736*K2**2*K3**2 - 152*K2**2*K4**2 + 1416*K2**2*K4 - 5346*K2**2 - 160*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1208*K2*K3*K5 + 144*K2*K4*K6 - 352*K3**4 - 144*K3**2*K4**2 + 296*K3**2*K6 - 3444*K3**2 + 72*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1422*K4**2 - 488*K5**2 - 70*K6**2 - 4*K7**2 - 2*K8**2 + 5646
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.208']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11007', 'vk6.11086', 'vk6.12175', 'vk6.12282', 'vk6.18209', 'vk6.18546', 'vk6.24669', 'vk6.25093', 'vk6.30570', 'vk6.30665', 'vk6.31842', 'vk6.31889', 'vk6.36797', 'vk6.37253', 'vk6.44038', 'vk6.44380', 'vk6.51818', 'vk6.51885', 'vk6.52684', 'vk6.52778', 'vk6.56003', 'vk6.56278', 'vk6.60540', 'vk6.60882', 'vk6.63496', 'vk6.63540', 'vk6.63976', 'vk6.64020', 'vk6.65662', 'vk6.65946', 'vk6.68708', 'vk6.68918']
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 O1O2O3O4O5U2O6U4U6U1U3U5
R3 orbit {'O1O2O3O4O5U2O6U4U6U1U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U3U5U6U2O6U4
Gauss code of K* O1O2O3O4O5U3U6U4U1U5O6U2
Gauss code of -K* O1O2O3O4O5U4O6U1U5U2U6U3
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 -3 1 -1 4 1],[ 2 0 -1 2 0 4 1],[ 3 1 0 2 1 3 1],[-1 -2 -2 0 -1 2 1],[ 1 0 -1 1 0 2 1],[-4 -4 -3 -2 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 4 1 1 -1 -2 -3],[-4 0 0 -2 -2 -4 -3],[-1 0 0 -1 -1 -1 -1],[-1 2 1 0 -1 -2 -2],[ 1 2 1 1 0 0 -1],[ 2 4 1 2 0 0 -1],[ 3 3 1 2 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,-1,1,2,3,0,2,2,4,3,1,1,1,1,1,2,2,0,1,1]
Phi over symmetry [-4,-1,-1,1,2,3,0,2,2,4,3,1,1,1,1,1,2,2,0,1,1]
Phi of -K [-3,-2,-1,1,1,4,0,1,2,3,4,1,1,2,2,1,1,3,-1,1,3]
Phi of K* [-4,-1,-1,1,2,3,1,3,3,2,4,1,1,1,2,1,2,3,1,1,0]
Phi of -K* [-3,-2,-1,1,1,4,1,1,1,2,3,0,1,2,4,1,1,2,-1,0,2]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t^2-t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+48t^4+19t^2+1
Outer characteristic polynomial t^7+80t^5+44t^3+5t
Flat arrow polynomial -6*K1**2 - 6*K1*K2 + 3*K1 - 2*K2**2 + 3*K2 + 3*K3 + K4 + 5
2-strand cable arrow polynomial 832*K1**4*K2 - 2384*K1**4 + 864*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1504*K1**3*K3 + 288*K1**2*K2**2*K4 - 3600*K1**2*K2**2 - 832*K1**2*K2*K4 + 7720*K1**2*K2 - 2224*K1**2*K3**2 - 256*K1**2*K4**2 - 6520*K1**2 + 192*K1*K2**3*K3 - 864*K1*K2**2*K3 - 384*K1*K2**2*K5 + 224*K1*K2*K3**3 + 96*K1*K2*K3*K4**2 - 832*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 8984*K1*K2*K3 + 3320*K1*K3*K4 + 728*K1*K4*K5 - 216*K2**4 - 736*K2**2*K3**2 - 152*K2**2*K4**2 + 1416*K2**2*K4 - 5346*K2**2 - 160*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1208*K2*K3*K5 + 144*K2*K4*K6 - 352*K3**4 - 144*K3**2*K4**2 + 296*K3**2*K6 - 3444*K3**2 + 72*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1422*K4**2 - 488*K5**2 - 70*K6**2 - 4*K7**2 - 2*K8**2 + 5646
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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