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Flat knot 6.1806

Min(phi) over symmetries of the knot is: [-1,-1,-1,1,1,1,-1,-1,1,1,1,0,1,1,1,1,1,2,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1741', '6.1806']
Arrow polynomial of the knot is: -8*K1**2 - 8*K1*K2 + 4*K1 + 4*K2 + 4*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.899', '6.912', '6.1806']
Outer characteristic polynomial of the knot is: t^7+17t^5+26t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1766', '6.1806']
2-strand cable arrow polynomial of the knot is: -256*K1**6 + 928*K1**4*K2 - 4000*K1**4 + 288*K1**3*K2*K3 + 64*K1**3*K3*K4 - 448*K1**3*K3 - 2736*K1**2*K2**2 - 288*K1**2*K2*K4 + 6896*K1**2*K2 - 1216*K1**2*K3**2 - 64*K1**2*K3*K5 - 336*K1**2*K4**2 - 3512*K1**2 - 704*K1*K2**2*K3 - 416*K1*K2*K3*K4 + 5096*K1*K2*K3 + 2256*K1*K3*K4 + 568*K1*K4*K5 - 96*K2**4 - 208*K2**2*K3**2 - 128*K2**2*K4**2 + 784*K2**2*K4 - 3536*K2**2 + 376*K2*K3*K5 + 128*K2*K4*K6 - 2064*K3**2 - 984*K4**2 - 224*K5**2 - 32*K6**2 + 3862
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1806']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.18884', 'vk6.18887', 'vk6.18888', 'vk6.18894', 'vk6.18897', 'vk6.18898', 'vk6.18960', 'vk6.18964', 'vk6.18965', 'vk6.18970', 'vk6.18974', 'vk6.18975', 'vk6.25587', 'vk6.25591', 'vk6.25592', 'vk6.25593', 'vk6.25597', 'vk6.25598', 'vk6.37614', 'vk6.37615', 'vk6.37619', 'vk6.37624', 'vk6.37625', 'vk6.37629', 'vk6.56414', 'vk6.56417', 'vk6.56449', 'vk6.56453', 'vk6.56454', 'vk6.56475', 'vk6.56479', 'vk6.56480']
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 O1O2O3U4U3O5O6U2U6O4U1U5
R3 orbit {'O1O2O3U4U3O5O6U2U6O4U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U3O5U6U2O6O4U1U5
Gauss code of K* O1O2U3O4O5U4U1U6O3O6U5U2
Gauss code of -K* O1O2U3O4O5U4U1O6O3U6U5U2
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 1 -1 1 1],[ 1 0 0 0 -1 1 1],[-1 -1 0 0 -1 -1 0],[ 1 1 1 1 0 1 1],[-1 -1 -1 1 -1 0 0],[-1 -1 -1 0 -1 0 0]]
Primitive based matrix [[ 0 1 1 1 -1 -1 -1],[-1 0 1 0 -1 -1 -1],[-1 -1 0 0 0 -1 -1],[-1 0 0 0 -1 -1 -1],[ 1 1 0 1 0 0 -1],[ 1 1 1 1 0 0 -1],[ 1 1 1 1 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,1,1,1,-1,0,1,1,1,0,0,1,1,1,1,1,0,1,1]
Phi over symmetry [-1,-1,-1,1,1,1,-1,-1,1,1,1,0,1,1,1,1,1,2,0,-1,0]
Phi of -K [-1,-1,-1,1,1,1,-1,-1,1,1,1,0,1,1,1,1,1,2,0,-1,0]
Phi of K* [-1,-1,-1,1,1,1,-1,0,1,1,2,0,1,1,1,1,1,1,-1,0,1]
Phi of -K* [-1,-1,-1,1,1,1,-1,0,0,1,1,1,1,1,1,1,1,1,-1,0,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 17z+35
Enhanced Jones-Krushkal polynomial 17w^2z+35w
Inner characteristic polynomial t^6+11t^4+12t^2
Outer characteristic polynomial t^7+17t^5+26t^3+4t
Flat arrow polynomial -8*K1**2 - 8*K1*K2 + 4*K1 + 4*K2 + 4*K3 + 5
2-strand cable arrow polynomial -256*K1**6 + 928*K1**4*K2 - 4000*K1**4 + 288*K1**3*K2*K3 + 64*K1**3*K3*K4 - 448*K1**3*K3 - 2736*K1**2*K2**2 - 288*K1**2*K2*K4 + 6896*K1**2*K2 - 1216*K1**2*K3**2 - 64*K1**2*K3*K5 - 336*K1**2*K4**2 - 3512*K1**2 - 704*K1*K2**2*K3 - 416*K1*K2*K3*K4 + 5096*K1*K2*K3 + 2256*K1*K3*K4 + 568*K1*K4*K5 - 96*K2**4 - 208*K2**2*K3**2 - 128*K2**2*K4**2 + 784*K2**2*K4 - 3536*K2**2 + 376*K2*K3*K5 + 128*K2*K4*K6 - 2064*K3**2 - 984*K4**2 - 224*K5**2 - 32*K6**2 + 3862
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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