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Flat knot 6.1356

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,0,1,1,3,3,0,0,1,1,0,-1,-1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1356']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 10*K1*K2 + 2*K1 + 6*K2 + 4*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.383', '6.922', '6.1172', '6.1356', '6.1359']
Outer characteristic polynomial of the knot is: t^7+39t^5+36t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1356']
2-strand cable arrow polynomial of the knot is: -512*K1**6 - 128*K1**4*K2**2 + 2272*K1**4*K2 - 6528*K1**4 + 800*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1664*K1**3*K3 + 384*K1**2*K2**3 - 5488*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 928*K1**2*K2*K4 + 12872*K1**2*K2 - 1760*K1**2*K3**2 - 64*K1**2*K3*K5 - 304*K1**2*K4**2 - 7520*K1**2 + 96*K1*K2**3*K3 - 896*K1*K2**2*K3 - 160*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 10048*K1*K2*K3 + 2992*K1*K3*K4 + 512*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 512*K2**4 - 32*K2**3*K6 - 432*K2**2*K3**2 - 168*K2**2*K4**2 + 1504*K2**2*K4 - 6856*K2**2 - 64*K2*K3**2*K4 + 680*K2*K3*K5 + 176*K2*K4*K6 - 32*K3**4 + 64*K3**2*K6 - 3728*K3**2 - 1368*K4**2 - 288*K5**2 - 56*K6**2 + 7246
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1356']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.18875', 'vk6.18877', 'vk6.18885', 'vk6.18891', 'vk6.18899', 'vk6.18901', 'vk6.18953', 'vk6.18955', 'vk6.18963', 'vk6.18969', 'vk6.18977', 'vk6.18979', 'vk6.25578', 'vk6.25580', 'vk6.25586', 'vk6.25596', 'vk6.25602', 'vk6.25604', 'vk6.37610', 'vk6.37612', 'vk6.37620', 'vk6.37626', 'vk6.37634', 'vk6.37636', 'vk6.56415', 'vk6.56420', 'vk6.56452', 'vk6.56458', 'vk6.56460', 'vk6.56466', 'vk6.56468', 'vk6.56474']
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 O1O2O3U2O4O5U1U5U3O6U4U6
R3 orbit {'O1O2O3U2O4O5U1U5U3O6U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4U1U6U3O6O5U2
Gauss code of K* O1O2O3U4O5O4U1U6U3O6U5U2
Gauss code of -K* O1O2O3U2U4O5U1U5U3O6O4U6
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 1 1 1 1],[ 3 0 0 3 3 1 1],[ 1 0 0 1 1 0 0],[-1 -3 -1 0 1 0 1],[-1 -3 -1 -1 0 0 1],[-1 -1 0 0 0 0 0],[-1 -1 0 -1 -1 0 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 1 1 0 -1 -3],[-1 -1 0 1 0 -1 -3],[-1 -1 -1 0 0 0 -1],[-1 0 0 0 0 0 -1],[ 1 1 1 0 0 0 0],[ 3 3 3 1 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,1,3,-1,-1,0,1,3,-1,0,1,3,0,0,1,0,1,0]
Phi over symmetry [-3,-1,1,1,1,1,0,1,1,3,3,0,0,1,1,0,-1,-1,0,0,-1]
Phi of -K [-3,-1,1,1,1,1,2,1,1,3,3,1,1,2,2,-1,-1,0,-1,0,0]
Phi of K* [-1,-1,-1,-1,1,3,-1,-1,0,2,3,-1,0,1,1,0,1,1,2,3,2]
Phi of -K* [-3,-1,1,1,1,1,0,1,1,3,3,0,0,1,1,0,-1,-1,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial z^2+22z+41
Enhanced Jones-Krushkal polynomial w^3z^2+22w^2z+41w
Inner characteristic polynomial t^6+25t^4+8t^2
Outer characteristic polynomial t^7+39t^5+36t^3+4t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 10*K1*K2 + 2*K1 + 6*K2 + 4*K3 + 7
2-strand cable arrow polynomial -512*K1**6 - 128*K1**4*K2**2 + 2272*K1**4*K2 - 6528*K1**4 + 800*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1664*K1**3*K3 + 384*K1**2*K2**3 - 5488*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 928*K1**2*K2*K4 + 12872*K1**2*K2 - 1760*K1**2*K3**2 - 64*K1**2*K3*K5 - 304*K1**2*K4**2 - 7520*K1**2 + 96*K1*K2**3*K3 - 896*K1*K2**2*K3 - 160*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 10048*K1*K2*K3 + 2992*K1*K3*K4 + 512*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 512*K2**4 - 32*K2**3*K6 - 432*K2**2*K3**2 - 168*K2**2*K4**2 + 1504*K2**2*K4 - 6856*K2**2 - 64*K2*K3**2*K4 + 680*K2*K3*K5 + 176*K2*K4*K6 - 32*K3**4 + 64*K3**2*K6 - 3728*K3**2 - 1368*K4**2 - 288*K5**2 - 56*K6**2 + 7246
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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