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Flat knot 6.1722

Min(phi) over symmetries of the knot is: [-1,-1,-1,1,1,1,-2,-1,0,1,2,-1,1,1,1,1,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1722']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1*K2 + K1 + 3*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1231', '6.1372', '6.1722', '6.1817', '6.1862', '6.2082']
Outer characteristic polynomial of the knot is: t^7+21t^5+47t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1722']
2-strand cable arrow polynomial of the knot is: 1504*K1**4*K2 - 3840*K1**4 + 928*K1**3*K2*K3 - 960*K1**3*K3 - 128*K1**2*K2**4 + 352*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4704*K1**2*K2**2 - 992*K1**2*K2*K4 + 6528*K1**2*K2 - 1280*K1**2*K3**2 - 64*K1**2*K4**2 - 3008*K1**2 + 352*K1*K2**3*K3 - 896*K1*K2**2*K3 - 192*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 6288*K1*K2*K3 + 2168*K1*K3*K4 + 168*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 464*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 1120*K2**2*K4 - 3178*K2**2 + 416*K2*K3*K5 + 104*K2*K4*K6 - 2104*K3**2 - 912*K4**2 - 152*K5**2 - 22*K6**2 + 3446
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1722']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10520', 'vk6.10524', 'vk6.10601', 'vk6.10609', 'vk6.10790', 'vk6.10798', 'vk6.10897', 'vk6.10901', 'vk6.19017', 'vk6.19033', 'vk6.19092', 'vk6.19094', 'vk6.19139', 'vk6.19141', 'vk6.25537', 'vk6.25553', 'vk6.25634', 'vk6.25650', 'vk6.25764', 'vk6.25766', 'vk6.30205', 'vk6.30209', 'vk6.30288', 'vk6.30296', 'vk6.30417', 'vk6.30425', 'vk6.37727', 'vk6.37743', 'vk6.56506', 'vk6.56514', 'vk6.66170', 'vk6.66178']
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 O1O2O3U2U1O4O5U4U3O6U5U6
R3 orbit {'O1O2O3U2U1O4O5U4U3O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4U1U6O5O6U3U2
Gauss code of K* O1O2U3O4O3U5U6U2O6O5U1U4
Gauss code of -K* O1O2U1O3O4U2U4O5O6U3U6U5
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 -1 -1 1 -1 1 1],[ 1 0 0 2 0 1 0],[ 1 0 0 1 0 1 0],[-1 -2 -1 0 0 2 1],[ 1 0 0 0 0 1 1],[-1 -1 -1 -2 -1 0 1],[-1 0 0 -1 -1 -1 0]]
Primitive based matrix [[ 0 1 1 1 -1 -1 -1],[-1 0 2 1 0 -1 -2],[-1 -2 0 1 -1 -1 -1],[-1 -1 -1 0 -1 0 0],[ 1 0 1 1 0 0 0],[ 1 1 1 0 0 0 0],[ 1 2 1 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,1,1,1,-2,-1,0,1,2,-1,1,1,1,1,0,0,0,0,0]
Phi over symmetry [-1,-1,-1,1,1,1,-2,-1,0,1,2,-1,1,1,1,1,0,0,0,0,0]
Phi of -K [-1,-1,-1,1,1,1,0,0,0,1,2,0,1,1,2,2,1,1,-2,-1,-1]
Phi of K* [-1,-1,-1,1,1,1,-2,1,1,1,1,1,0,1,2,2,2,1,0,0,0]
Phi of -K* [-1,-1,-1,1,1,1,0,0,0,1,1,0,0,1,2,1,1,0,-1,-1,-2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+15t^4+15t^2+1
Outer characteristic polynomial t^7+21t^5+47t^3+7t
Flat arrow polynomial 4*K1**3 - 8*K1*K2 + K1 + 3*K3 + 1
2-strand cable arrow polynomial 1504*K1**4*K2 - 3840*K1**4 + 928*K1**3*K2*K3 - 960*K1**3*K3 - 128*K1**2*K2**4 + 352*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4704*K1**2*K2**2 - 992*K1**2*K2*K4 + 6528*K1**2*K2 - 1280*K1**2*K3**2 - 64*K1**2*K4**2 - 3008*K1**2 + 352*K1*K2**3*K3 - 896*K1*K2**2*K3 - 192*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 6288*K1*K2*K3 + 2168*K1*K3*K4 + 168*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 464*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 1120*K2**2*K4 - 3178*K2**2 + 416*K2*K3*K5 + 104*K2*K4*K6 - 2104*K3**2 - 912*K4**2 - 152*K5**2 - 22*K6**2 + 3446
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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