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Flat knot 6.1817

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,1,2,-1,0,1,2,0,1,0,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1817']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1*K2 + K1 + 3*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1231', '6.1372', '6.1722', '6.1817', '6.1862', '6.2082']
Outer characteristic polynomial of the knot is: t^7+20t^5+43t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1817']
2-strand cable arrow polynomial of the knot is: 1984*K1**4*K2 - 3584*K1**4 + 512*K1**3*K2*K3 - 1376*K1**3*K3 - 128*K1**2*K2**4 + 448*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4800*K1**2*K2**2 - 832*K1**2*K2*K4 + 6336*K1**2*K2 - 672*K1**2*K3**2 - 32*K1**2*K4**2 - 2680*K1**2 + 384*K1*K2**3*K3 - 512*K1*K2**2*K3 - 288*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 5512*K1*K2*K3 + 1168*K1*K3*K4 + 200*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 528*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 976*K2**2*K4 - 2626*K2**2 + 440*K2*K3*K5 + 104*K2*K4*K6 - 1456*K3**2 - 560*K4**2 - 144*K5**2 - 22*K6**2 + 2750
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1817']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20144', 'vk6.20150', 'vk6.20160', 'vk6.20166', 'vk6.21436', 'vk6.21448', 'vk6.27260', 'vk6.27270', 'vk6.27288', 'vk6.27290', 'vk6.28922', 'vk6.28932', 'vk6.28950', 'vk6.38683', 'vk6.38689', 'vk6.38709', 'vk6.38723', 'vk6.40889', 'vk6.40903', 'vk6.47269', 'vk6.47284', 'vk6.47290', 'vk6.56969', 'vk6.56983', 'vk6.56988', 'vk6.57008', 'vk6.58123', 'vk6.62670', 'vk6.62690', 'vk6.67462', 'vk6.70034', 'vk6.70046']
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 O1O2O3U4U5O6O5U3U2O4U1U6
R3 orbit {'O1O2O3U4U5O6O5U3U2O4U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U3O5U2U1O6O4U6U5
Gauss code of K* O1O2U3O4O5U4U2U1O3O6U5U6
Gauss code of -K* O1O2U3O4O5U6U1O6O3U5U4U2
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 0 0 -1 1 1],[ 1 0 1 1 -1 1 1],[ 0 -1 0 0 -1 0 0],[ 0 -1 0 0 0 0 -1],[ 1 1 1 0 0 2 2],[-1 -1 0 0 -2 0 1],[-1 -1 0 1 -2 -1 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 1 0 0 -1 -2],[-1 -1 0 1 0 -1 -2],[ 0 0 -1 0 0 -1 0],[ 0 0 0 0 0 -1 -1],[ 1 1 1 1 1 0 -1],[ 1 2 2 0 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,-1,0,0,1,2,-1,0,1,2,0,1,0,1,1,1]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,0,1,2,-1,0,1,2,0,1,0,1,1,1]
Phi of -K [-1,-1,0,0,1,1,-1,0,1,0,0,0,0,1,1,0,1,1,1,2,-1]
Phi of K* [-1,-1,0,0,1,1,-1,1,2,0,1,1,1,0,1,0,0,0,1,0,1]
Phi of -K* [-1,-1,0,0,1,1,-1,1,1,1,1,0,1,2,2,0,-1,0,0,0,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z^2+26z+25
Enhanced Jones-Krushkal polynomial 7w^3z^2+26w^2z+25w
Inner characteristic polynomial t^6+16t^4+21t^2
Outer characteristic polynomial t^7+20t^5+43t^3+5t
Flat arrow polynomial 4*K1**3 - 8*K1*K2 + K1 + 3*K3 + 1
2-strand cable arrow polynomial 1984*K1**4*K2 - 3584*K1**4 + 512*K1**3*K2*K3 - 1376*K1**3*K3 - 128*K1**2*K2**4 + 448*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4800*K1**2*K2**2 - 832*K1**2*K2*K4 + 6336*K1**2*K2 - 672*K1**2*K3**2 - 32*K1**2*K4**2 - 2680*K1**2 + 384*K1*K2**3*K3 - 512*K1*K2**2*K3 - 288*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 5512*K1*K2*K3 + 1168*K1*K3*K4 + 200*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 528*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 976*K2**2*K4 - 2626*K2**2 + 440*K2*K3*K5 + 104*K2*K4*K6 - 1456*K3**2 - 560*K4**2 - 144*K5**2 - 22*K6**2 + 2750
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}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}]]
If K is slice False
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