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Flat knot 6.441

Min(phi) over symmetries of the knot is: [-4,-1,1,1,1,2,1,2,4,4,2,1,1,2,1,-1,0,0,0,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.441']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.96', '6.149', '6.269', '6.441', '6.457']
Outer characteristic polynomial of the knot is: t^7+78t^5+117t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.441']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 608*K1**4*K2 - 1072*K1**4 + 288*K1**3*K2*K3 - 416*K1**3*K3 - 256*K1**2*K2**4 + 704*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 2592*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 160*K1**2*K2*K4 + 3600*K1**2*K2 - 368*K1**2*K3**2 - 2412*K1**2 + 576*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 736*K1*K2**2*K3 - 160*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3256*K1*K2*K3 + 584*K1*K3*K4 + 56*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 640*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 448*K2**2*K3**2 - 32*K2**2*K3*K7 - 112*K2**2*K4**2 + 792*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 1810*K2**2 + 368*K2*K3*K5 + 64*K2*K4*K6 + 8*K2*K5*K7 + 24*K3**2*K6 - 1044*K3**2 - 314*K4**2 - 80*K5**2 - 22*K6**2 + 2016
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.441']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16535', 'vk6.16628', 'vk6.16962', 'vk6.17204', 'vk6.17528', 'vk6.17584', 'vk6.18852', 'vk6.18931', 'vk6.19213', 'vk6.19506', 'vk6.22245', 'vk6.23061', 'vk6.24132', 'vk6.25482', 'vk6.26020', 'vk6.26404', 'vk6.28305', 'vk6.34930', 'vk6.35046', 'vk6.35411', 'vk6.35833', 'vk6.35843', 'vk6.36315', 'vk6.36389', 'vk6.37579', 'vk6.39911', 'vk6.39932', 'vk6.42502', 'vk6.42614', 'vk6.43160', 'vk6.43493', 'vk6.44609', 'vk6.46465', 'vk6.54764', 'vk6.55122', 'vk6.55376', 'vk6.56559', 'vk6.59817', 'vk6.60192', 'vk6.66092']
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 O1O2O3O4O5U6U2O6U3U4U1U5
R3 orbit {'O1O2O3O4O5U6U2O6U3U4U1U5', 'O1O2O3O4O5U3U6U2O6U4U1U5'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U1U5U2U3O6U4U6
Gauss code of K* O1O2O3O4U3U5U1U2U4O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U1U3U4U6U2
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 4 -1],[ 1 0 -1 0 2 4 0],[ 2 1 0 0 1 2 2],[ 1 0 0 0 1 2 1],[-1 -2 -1 -1 0 1 -1],[-4 -4 -2 -2 -1 0 -4],[ 1 0 -2 -1 1 4 0]]
Primitive based matrix [[ 0 4 1 -1 -1 -1 -2],[-4 0 -1 -2 -4 -4 -2],[-1 1 0 -1 -1 -2 -1],[ 1 2 1 0 1 0 0],[ 1 4 1 -1 0 0 -2],[ 1 4 2 0 0 0 -1],[ 2 2 1 0 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,1,1,1,2,1,2,4,4,2,1,1,2,1,-1,0,0,0,2,1]
Phi over symmetry [-4,-1,1,1,1,2,1,2,4,4,2,1,1,2,1,-1,0,0,0,2,1]
Phi of -K [-2,-1,-1,-1,1,4,-1,0,1,2,4,0,1,1,1,0,0,1,1,3,2]
Phi of K* [-4,-1,1,1,1,2,2,1,1,3,4,0,1,1,2,0,0,0,-1,-1,1]
Phi of -K* [-2,-1,-1,-1,1,4,0,1,2,1,2,0,1,1,2,0,2,4,1,4,1]
Symmetry type of based matrix c
u-polynomial -t^4+t^2+2t
Normalized Jones-Krushkal polynomial 2z^2+15z+23
Enhanced Jones-Krushkal polynomial 2w^3z^2+15w^2z+23w
Inner characteristic polynomial t^6+54t^4+74t^2+1
Outer characteristic polynomial t^7+78t^5+117t^3+4t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -256*K1**4*K2**2 + 608*K1**4*K2 - 1072*K1**4 + 288*K1**3*K2*K3 - 416*K1**3*K3 - 256*K1**2*K2**4 + 704*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 2592*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 160*K1**2*K2*K4 + 3600*K1**2*K2 - 368*K1**2*K3**2 - 2412*K1**2 + 576*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 736*K1*K2**2*K3 - 160*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3256*K1*K2*K3 + 584*K1*K3*K4 + 56*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 640*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 448*K2**2*K3**2 - 32*K2**2*K3*K7 - 112*K2**2*K4**2 + 792*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 1810*K2**2 + 368*K2*K3*K5 + 64*K2*K4*K6 + 8*K2*K5*K7 + 24*K3**2*K6 - 1044*K3**2 - 314*K4**2 - 80*K5**2 - 22*K6**2 + 2016
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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