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Flat knot 6.649

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,1,1,3,3,0,0,1,1,0,0,0,0,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.649']
Arrow polynomial of the knot is: -10*K1**2 - 2*K1*K2 + K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.574', '6.593', '6.604', '6.649', '6.650', '6.673', '6.690', '6.783', '6.973', '6.985', '6.1033', '6.1035']
Outer characteristic polynomial of the knot is: t^7+58t^5+49t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.649']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 384*K1**4*K2**2 + 1824*K1**4*K2 - 5088*K1**4 + 352*K1**3*K2*K3 - 288*K1**3*K3 + 1248*K1**2*K2**3 - 7744*K1**2*K2**2 - 416*K1**2*K2*K4 + 9728*K1**2*K2 - 480*K1**2*K3**2 - 96*K1**2*K3*K5 - 3020*K1**2 - 1280*K1*K2**2*K3 + 6576*K1*K2*K3 + 976*K1*K3*K4 + 64*K1*K4*K5 - 1592*K2**4 - 208*K2**2*K3**2 - 8*K2**2*K4**2 + 1536*K2**2*K4 - 3134*K2**2 + 240*K2*K3*K5 + 8*K2*K4*K6 - 1544*K3**2 - 474*K4**2 - 76*K5**2 - 2*K6**2 + 3664
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.649']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11568', 'vk6.11577', 'vk6.11909', 'vk6.11917', 'vk6.12919', 'vk6.13227', 'vk6.13232', 'vk6.20955', 'vk6.20965', 'vk6.22372', 'vk6.22381', 'vk6.28422', 'vk6.31353', 'vk6.31363', 'vk6.31763', 'vk6.32519', 'vk6.32525', 'vk6.32920', 'vk6.32926', 'vk6.40136', 'vk6.40140', 'vk6.42146', 'vk6.46650', 'vk6.46653', 'vk6.52350', 'vk6.52613', 'vk6.52617', 'vk6.53482', 'vk6.53489', 'vk6.58947', 'vk6.64479', 'vk6.69783']
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 O1O2O3O4U1O5U6U4O6U3U5U2
R3 orbit {'O1O2O3O4U1O5U6U4O6U3U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U2O6U1U6O5U4
Gauss code of K* O1O2O3U4U3U1U5O4U2O6O5U6
Gauss code of -K* O1O2O3U4O5O4U2O6U5U3U1U6
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 -3 1 0 1 2 -1],[ 3 0 3 2 1 2 2],[-1 -3 0 -1 0 2 -2],[ 0 -2 1 0 1 2 -1],[-1 -1 0 -1 0 0 -1],[-2 -2 -2 -2 0 0 -2],[ 1 -2 2 1 1 2 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -2 -2 -2 -2],[-1 0 0 0 -1 -1 -1],[-1 2 0 0 -1 -2 -3],[ 0 2 1 1 0 -1 -2],[ 1 2 1 2 1 0 -2],[ 3 2 1 3 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,2,2,2,2,0,1,1,1,1,2,3,1,2,2]
Phi over symmetry [-3,-1,0,1,1,2,0,1,1,3,3,0,0,1,1,0,0,0,0,-1,1]
Phi of -K [-3,-1,0,1,1,2,0,1,1,3,3,0,0,1,1,0,0,0,0,-1,1]
Phi of K* [-2,-1,-1,0,1,3,-1,1,0,1,3,0,0,0,1,0,1,3,0,1,0]
Phi of -K* [-3,-1,0,1,1,2,2,2,1,3,2,1,1,2,2,1,1,2,0,0,2]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+42t^4+28t^2+4
Outer characteristic polynomial t^7+58t^5+49t^3+9t
Flat arrow polynomial -10*K1**2 - 2*K1*K2 + K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -384*K1**6 - 384*K1**4*K2**2 + 1824*K1**4*K2 - 5088*K1**4 + 352*K1**3*K2*K3 - 288*K1**3*K3 + 1248*K1**2*K2**3 - 7744*K1**2*K2**2 - 416*K1**2*K2*K4 + 9728*K1**2*K2 - 480*K1**2*K3**2 - 96*K1**2*K3*K5 - 3020*K1**2 - 1280*K1*K2**2*K3 + 6576*K1*K2*K3 + 976*K1*K3*K4 + 64*K1*K4*K5 - 1592*K2**4 - 208*K2**2*K3**2 - 8*K2**2*K4**2 + 1536*K2**2*K4 - 3134*K2**2 + 240*K2*K3*K5 + 8*K2*K4*K6 - 1544*K3**2 - 474*K4**2 - 76*K5**2 - 2*K6**2 + 3664
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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