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Flat knot 6.1867

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,1,0,1,0,0,1,1,-1,2,0,1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1867']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.414', '6.594', '6.608', '6.790', '6.1233', '6.1285', '6.1293', '6.1513', '6.1752', '6.1787', '6.1810', '6.1818', '6.1867', '6.1868', '6.1923']
Outer characteristic polynomial of the knot is: t^7+20t^5+89t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1867']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 256*K1**4*K2**3 - 1536*K1**4*K2**2 + 3008*K1**4*K2 - 4192*K1**4 - 256*K1**3*K2**2*K3 + 1216*K1**3*K2*K3 - 928*K1**3*K3 - 640*K1**2*K2**4 + 3712*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 10848*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 928*K1**2*K2*K4 + 10528*K1**2*K2 - 416*K1**2*K3**2 - 32*K1**2*K4**2 - 4568*K1**2 + 1248*K1*K2**3*K3 - 2336*K1*K2**2*K3 - 384*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 8344*K1*K2*K3 + 776*K1*K3*K4 + 96*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 2784*K2**4 - 64*K2**3*K6 - 832*K2**2*K3**2 - 128*K2**2*K4**2 + 2432*K2**2*K4 - 3404*K2**2 + 648*K2*K3*K5 + 80*K2*K4*K6 - 1752*K3**2 - 528*K4**2 - 152*K5**2 - 12*K6**2 + 4278
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1867']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19931', 'vk6.19988', 'vk6.21166', 'vk6.21253', 'vk6.26882', 'vk6.27015', 'vk6.28642', 'vk6.28734', 'vk6.38307', 'vk6.38423', 'vk6.40436', 'vk6.40600', 'vk6.45175', 'vk6.45307', 'vk6.47011', 'vk6.47084', 'vk6.56717', 'vk6.56794', 'vk6.57806', 'vk6.57926', 'vk6.61135', 'vk6.61294', 'vk6.62382', 'vk6.62482', 'vk6.66410', 'vk6.66502', 'vk6.67172', 'vk6.67290', 'vk6.69061', 'vk6.69152', 'vk6.69848', 'vk6.69908']
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 O1O2O3U4U5O6U1O4O5U3U6U2
R3 orbit {'O1O2O3U4U5O6U1O4O5U3U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4U1O5O6U3O4U5U6
Gauss code of K* O1O2O3U4U3U1O5O6U2O4U5U6
Gauss code of -K* O1O2O3U4U5O6U2O4O5U3U1U6
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 0 -1 1 0],[ 1 0 1 -1 1 2 0],[-1 -1 0 -1 -1 1 -1],[ 0 1 1 0 -1 1 -1],[ 1 -1 1 1 0 1 1],[-1 -2 -1 -1 -1 0 0],[ 0 0 1 1 -1 0 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 1 -1 -1 -1 -1],[-1 -1 0 0 -1 -1 -2],[ 0 1 0 0 1 -1 0],[ 0 1 1 -1 0 -1 1],[ 1 1 1 1 1 0 -1],[ 1 1 2 0 -1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,-1,1,1,1,1,0,1,1,2,-1,1,0,1,-1,1]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,1,0,1,0,0,1,1,-1,2,0,1,0,1]
Phi of -K [-1,-1,0,0,1,1,-1,1,2,0,1,0,0,1,1,-1,1,0,0,0,1]
Phi of K* [-1,-1,0,0,1,1,-1,0,1,0,1,0,0,1,1,-1,2,0,1,0,1]
Phi of -K* [-1,-1,0,0,1,1,-1,1,1,1,1,-1,0,1,2,-1,1,1,1,0,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+16t^4+59t^2+4
Outer characteristic polynomial t^7+20t^5+89t^3+10t
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -128*K1**6 + 256*K1**4*K2**3 - 1536*K1**4*K2**2 + 3008*K1**4*K2 - 4192*K1**4 - 256*K1**3*K2**2*K3 + 1216*K1**3*K2*K3 - 928*K1**3*K3 - 640*K1**2*K2**4 + 3712*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 10848*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 928*K1**2*K2*K4 + 10528*K1**2*K2 - 416*K1**2*K3**2 - 32*K1**2*K4**2 - 4568*K1**2 + 1248*K1*K2**3*K3 - 2336*K1*K2**2*K3 - 384*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 8344*K1*K2*K3 + 776*K1*K3*K4 + 96*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 2784*K2**4 - 64*K2**3*K6 - 832*K2**2*K3**2 - 128*K2**2*K4**2 + 2432*K2**2*K4 - 3404*K2**2 + 648*K2*K3*K5 + 80*K2*K4*K6 - 1752*K3**2 - 528*K4**2 - 152*K5**2 - 12*K6**2 + 4278
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}]]
If K is slice False
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