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Flat knot 6.1741

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: -12*K1**2 - 8*K1*K2 + 4*K1 + 6*K2 + 4*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.763', '6.1515', '6.1741', '6.1825']
Outer characteristic polynomial of the knot is: t^7+21t^5+22t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1741']
2-strand cable arrow polynomial of the knot is: -768*K1**6 - 384*K1**4*K2**2 + 2592*K1**4*K2 - 6528*K1**4 + 1088*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1664*K1**3*K3 - 5168*K1**2*K2**2 - 832*K1**2*K2*K4 + 12032*K1**2*K2 - 1248*K1**2*K3**2 - 32*K1**2*K3*K5 - 208*K1**2*K4**2 - 5848*K1**2 - 384*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 7800*K1*K2*K3 + 2080*K1*K3*K4 + 304*K1*K4*K5 - 240*K2**4 - 272*K2**2*K3**2 - 128*K2**2*K4**2 + 840*K2**2*K4 - 5424*K2**2 + 328*K2*K3*K5 + 128*K2*K4*K6 - 2632*K3**2 - 908*K4**2 - 160*K5**2 - 32*K6**2 + 5762
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1741']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10956', 'vk6.10957', 'vk6.10988', 'vk6.10989', 'vk6.12122', 'vk6.12123', 'vk6.12154', 'vk6.12155', 'vk6.13790', 'vk6.13799', 'vk6.14214', 'vk6.14222', 'vk6.14663', 'vk6.14671', 'vk6.14861', 'vk6.14874', 'vk6.15817', 'vk6.15825', 'vk6.31829', 'vk6.31830', 'vk6.33628', 'vk6.33631', 'vk6.33661', 'vk6.33662', 'vk6.51801', 'vk6.51802', 'vk6.52663', 'vk6.52664', 'vk6.53800', 'vk6.53825', 'vk6.54224', 'vk6.54232']
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 O1O2O3U4U1O4O5U2U5O6U3U6
R3 orbit {'O1O2O3U4U1O4O5U2U5O6U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U1O4U5U2O5O6U3U6
Gauss code of K* O1O2U3O4O3U5U1U4O6O5U6U2
Gauss code of -K* O1O2U1O3O4U3U5O6O5U2U4U6
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 2 1 1 1],[-1 -1 -2 0 -1 0 1],[ 1 -1 -1 1 0 1 1],[-1 -1 -1 0 -1 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 1 1 1 -1 -1 -1],[-1 0 1 0 -1 -1 -2],[-1 -1 0 0 -1 -1 -1],[-1 0 0 0 -1 -1 -1],[ 1 1 1 1 0 1 0],[ 1 1 1 1 -1 0 -1],[ 1 2 1 1 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,1,1,1,-1,0,1,1,2,0,1,1,1,1,1,1,-1,0,1]
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,-1,0,0,1,1,1,1,1,1,1,1,1,-1,0,0]
Phi of K* [-1,-1,-1,1,1,1,-1,0,1,1,1,0,0,1,1,1,1,1,0,1,1]
Phi of -K* [-1,-1,-1,1,1,1,-1,-1,1,1,1,0,1,1,1,1,1,2,0,-1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial z^2+22z+41
Enhanced Jones-Krushkal polynomial w^3z^2+22w^2z+41w
Inner characteristic polynomial t^6+15t^4+8t^2
Outer characteristic polynomial t^7+21t^5+22t^3+4t
Flat arrow polynomial -12*K1**2 - 8*K1*K2 + 4*K1 + 6*K2 + 4*K3 + 7
2-strand cable arrow polynomial -768*K1**6 - 384*K1**4*K2**2 + 2592*K1**4*K2 - 6528*K1**4 + 1088*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1664*K1**3*K3 - 5168*K1**2*K2**2 - 832*K1**2*K2*K4 + 12032*K1**2*K2 - 1248*K1**2*K3**2 - 32*K1**2*K3*K5 - 208*K1**2*K4**2 - 5848*K1**2 - 384*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 7800*K1*K2*K3 + 2080*K1*K3*K4 + 304*K1*K4*K5 - 240*K2**4 - 272*K2**2*K3**2 - 128*K2**2*K4**2 + 840*K2**2*K4 - 5424*K2**2 + 328*K2*K3*K5 + 128*K2*K4*K6 - 2632*K3**2 - 908*K4**2 - 160*K5**2 - 32*K6**2 + 5762
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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