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Flat knot 6.927

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,1,2,3,1,1,1,1,0,1,1,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.927']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.927', '6.1364', '6.1367', '6.1540', '6.1675', '6.1779', '6.1811', '6.1876', '6.2075']
Outer characteristic polynomial of the knot is: t^7+40t^5+61t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.927']
2-strand cable arrow polynomial of the knot is: 1888*K1**4*K2 - 4352*K1**4 + 832*K1**3*K2*K3 - 928*K1**3*K3 - 128*K1**2*K2**4 + 1312*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8064*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 9632*K1**2*K2 - 1248*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 4104*K1**2 + 480*K1*K2**3*K3 - 1888*K1*K2**2*K3 - 224*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 8168*K1*K2*K3 + 1568*K1*K3*K4 + 96*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 1488*K2**4 - 64*K2**3*K6 - 608*K2**2*K3**2 - 128*K2**2*K4**2 + 1744*K2**2*K4 - 3724*K2**2 + 536*K2*K3*K5 + 80*K2*K4*K6 - 2008*K3**2 - 572*K4**2 - 96*K5**2 - 12*K6**2 + 4034
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.927']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13906', 'vk6.14001', 'vk6.14172', 'vk6.14413', 'vk6.14977', 'vk6.15098', 'vk6.15644', 'vk6.16100', 'vk6.16712', 'vk6.16740', 'vk6.16854', 'vk6.18815', 'vk6.19279', 'vk6.19573', 'vk6.23153', 'vk6.23237', 'vk6.25413', 'vk6.26466', 'vk6.33725', 'vk6.33800', 'vk6.34279', 'vk6.35145', 'vk6.37534', 'vk6.42742', 'vk6.44684', 'vk6.54111', 'vk6.54929', 'vk6.54959', 'vk6.56401', 'vk6.56604', 'vk6.59356', 'vk6.64589']
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 O1O2O3O4U3U5O6O5U2U4U1U6
R3 orbit {'O1O2O3O4U3U5O6O5U2U4U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4U1U3O6O5U6U2
Gauss code of K* O1O2O3O4U3U1U5U2O5O6U4U6
Gauss code of -K* O1O2O3O4U5U1O5O6U3U6U4U2
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 -2 -1 1 1 2],[ 1 0 -1 -1 2 1 2],[ 2 1 0 0 2 2 1],[ 1 1 0 0 1 1 1],[-1 -2 -2 -1 0 -1 0],[-1 -1 -2 -1 1 0 2],[-2 -2 -1 -1 0 -2 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -2 -1 -2 -1],[-1 0 0 -1 -1 -2 -2],[-1 2 1 0 -1 -1 -2],[ 1 1 1 1 0 1 0],[ 1 2 2 1 -1 0 -1],[ 2 1 2 2 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,2,1,2,1,1,1,2,2,1,1,2,-1,0,1]
Phi over symmetry [-2,-1,-1,1,1,2,-1,1,1,2,3,1,1,1,1,0,1,1,-1,0,1]
Phi of -K [-2,-1,-1,1,1,2,0,1,1,1,3,1,0,1,1,1,1,2,1,1,-1]
Phi of K* [-2,-1,-1,1,1,2,-1,1,1,2,3,1,1,1,1,0,1,1,-1,0,1]
Phi of -K* [-2,-1,-1,1,1,2,0,1,2,2,1,1,1,1,1,1,2,2,1,2,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z^2+28z+29
Enhanced Jones-Krushkal polynomial 7w^3z^2+28w^2z+29w
Inner characteristic polynomial t^6+28t^4+25t^2+1
Outer characteristic polynomial t^7+40t^5+61t^3+8t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial 1888*K1**4*K2 - 4352*K1**4 + 832*K1**3*K2*K3 - 928*K1**3*K3 - 128*K1**2*K2**4 + 1312*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8064*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 9632*K1**2*K2 - 1248*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 4104*K1**2 + 480*K1*K2**3*K3 - 1888*K1*K2**2*K3 - 224*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 8168*K1*K2*K3 + 1568*K1*K3*K4 + 96*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 1488*K2**4 - 64*K2**3*K6 - 608*K2**2*K3**2 - 128*K2**2*K4**2 + 1744*K2**2*K4 - 3724*K2**2 + 536*K2*K3*K5 + 80*K2*K4*K6 - 2008*K3**2 - 572*K4**2 - 96*K5**2 - 12*K6**2 + 4034
Genus of based matrix 1
Fillings of based matrix [[{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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