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Flat knot 6.1568

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,2,2,1,1,1,1,0,0,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1568']
Arrow polynomial of the knot is: -6*K1**2 - 8*K1*K2 + 4*K1 + 3*K2 + 4*K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1568']
Outer characteristic polynomial of the knot is: t^7+22t^5+26t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1568']
2-strand cable arrow polynomial of the knot is: 1792*K1**4*K2 - 3808*K1**4 + 864*K1**3*K2*K3 - 1792*K1**3*K3 + 288*K1**2*K2**2*K4 - 4112*K1**2*K2**2 - 992*K1**2*K2*K4 + 8544*K1**2*K2 - 1856*K1**2*K3**2 - 128*K1**2*K4**2 - 5644*K1**2 + 192*K1*K2**3*K3 - 864*K1*K2**2*K3 - 256*K1*K2**2*K5 - 608*K1*K2*K3*K4 + 7984*K1*K2*K3 + 2784*K1*K3*K4 + 360*K1*K4*K5 - 216*K2**4 - 288*K2**2*K3**2 - 128*K2**2*K4**2 + 1352*K2**2*K4 - 4920*K2**2 + 608*K2*K3*K5 + 128*K2*K4*K6 - 2844*K3**2 - 1190*K4**2 - 224*K5**2 - 32*K6**2 + 5004
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1568']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81571', 'vk6.81576', 'vk6.81652', 'vk6.81657', 'vk6.81735', 'vk6.81736', 'vk6.81855', 'vk6.81857', 'vk6.82237', 'vk6.82242', 'vk6.82393', 'vk6.82394', 'vk6.82508', 'vk6.82509', 'vk6.82571', 'vk6.82573', 'vk6.83169', 'vk6.83177', 'vk6.83592', 'vk6.83607', 'vk6.84137', 'vk6.84140', 'vk6.84338', 'vk6.84339', 'vk6.84555', 'vk6.84557', 'vk6.86482', 'vk6.86498', 'vk6.88730', 'vk6.88731', 'vk6.88914', 'vk6.88916']
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 O1O2O3U4O5U1U3O6O4U2U5U6
R3 orbit {'O1O2O3U4O5U1U3O6O4U2U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5U2O6O4U1U3O5U6
Gauss code of K* O1O2O3U4U1U5O6U2O4O5U3U6
Gauss code of -K* O1O2O3U4U1O5O6U2O4U5U3U6
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 -1 1 0 1 1],[ 2 0 1 1 1 2 1],[ 1 -1 0 1 0 1 1],[-1 -1 -1 0 -1 0 0],[ 0 -1 0 1 0 1 0],[-1 -2 -1 0 -1 0 1],[-1 -1 -1 0 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -2],[-1 -1 0 0 0 -1 -1],[-1 0 0 0 -1 -1 -1],[ 0 1 0 1 0 0 -1],[ 1 1 1 1 0 0 -1],[ 2 2 1 1 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,0,1,1,2,0,0,1,1,1,1,1,0,1,1]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,2,2,1,1,1,1,0,0,1,0,-1,0]
Phi of -K [-2,-1,0,1,1,1,0,1,1,2,2,1,1,1,1,0,0,1,0,-1,0]
Phi of K* [-1,-1,-1,0,1,2,-1,0,1,1,2,0,0,1,1,0,1,2,1,1,0]
Phi of -K* [-2,-1,0,1,1,1,1,1,1,1,2,0,1,1,1,0,1,1,0,-1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 4z^2+25z+35
Enhanced Jones-Krushkal polynomial 4w^3z^2+25w^2z+35w
Inner characteristic polynomial t^6+14t^4+13t^2+1
Outer characteristic polynomial t^7+22t^5+26t^3+4t
Flat arrow polynomial -6*K1**2 - 8*K1*K2 + 4*K1 + 3*K2 + 4*K3 + 4
2-strand cable arrow polynomial 1792*K1**4*K2 - 3808*K1**4 + 864*K1**3*K2*K3 - 1792*K1**3*K3 + 288*K1**2*K2**2*K4 - 4112*K1**2*K2**2 - 992*K1**2*K2*K4 + 8544*K1**2*K2 - 1856*K1**2*K3**2 - 128*K1**2*K4**2 - 5644*K1**2 + 192*K1*K2**3*K3 - 864*K1*K2**2*K3 - 256*K1*K2**2*K5 - 608*K1*K2*K3*K4 + 7984*K1*K2*K3 + 2784*K1*K3*K4 + 360*K1*K4*K5 - 216*K2**4 - 288*K2**2*K3**2 - 128*K2**2*K4**2 + 1352*K2**2*K4 - 4920*K2**2 + 608*K2*K3*K5 + 128*K2*K4*K6 - 2844*K3**2 - 1190*K4**2 - 224*K5**2 - 32*K6**2 + 5004
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {1, 3}]]
If K is slice False
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