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Flat knot 6.407

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,-1,1,1,2,3,0,1,1,1,0,0,0,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.407', '7.20043']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.115', '6.407', '6.413', '6.448', '6.844', '6.879', '6.888', '6.926', '6.934', '6.1140', '6.1143', '6.1161', '6.1177']
Outer characteristic polynomial of the knot is: t^7+35t^5+52t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.407', '7.20043']
2-strand cable arrow polynomial of the knot is: -3136*K1**4 + 1920*K1**3*K2*K3 + 128*K1**3*K3*K4 - 896*K1**3*K3 + 640*K1**2*K2**2*K4 - 6352*K1**2*K2**2 - 1184*K1**2*K2*K4 + 6912*K1**2*K2 - 1792*K1**2*K3**2 - 192*K1**2*K3*K5 - 416*K1**2*K4**2 - 1868*K1**2 + 1952*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 1248*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 800*K1*K2**2*K5 - 672*K1*K2*K3*K4 - 224*K1*K2*K3*K6 + 6560*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 1984*K1*K3*K4 + 456*K1*K4*K5 + 24*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4*K4**2 + 736*K2**4*K4 - 2304*K2**4 + 64*K2**3*K4*K6 - 256*K2**3*K6 - 1472*K2**2*K3**2 - 776*K2**2*K4**2 + 2016*K2**2*K4 - 16*K2**2*K5**2 - 48*K2**2*K6**2 - 1566*K2**2 + 984*K2*K3*K5 + 448*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 1464*K3**2 - 648*K4**2 - 160*K5**2 - 58*K6**2 - 4*K7**2 - 2*K8**2 + 2408
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.407']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.51', 'vk6.102', 'vk6.197', 'vk6.250', 'vk6.387', 'vk6.794', 'vk6.803', 'vk6.1246', 'vk6.1335', 'vk6.1390', 'vk6.1544', 'vk6.2021', 'vk6.2416', 'vk6.2433', 'vk6.2672', 'vk6.2977', 'vk6.10453', 'vk6.10470', 'vk6.10674', 'vk6.10861', 'vk6.14650', 'vk6.16259', 'vk6.19177', 'vk6.25739', 'vk6.25904', 'vk6.30148', 'vk6.30357', 'vk6.30484', 'vk6.33413', 'vk6.33567', 'vk6.53785', 'vk6.63416']
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 O1O2O3O4O5U4U5O6U1U6U3U2
R3 orbit {'O1O2O3O4O5U4U5O6U1U6U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U3U6U5O6U1U2
Gauss code of K* O1O2O3O4U1U4U3U5U6O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U5U6U2U1U4
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 3 2 -1 1 1],[-1 -3 0 0 -1 1 0],[-1 -2 0 0 -1 1 0],[ 1 1 1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 1 0 0 -1 -2],[-1 -1 0 0 -1 -1 -1],[-1 0 0 0 0 0 -1],[-1 0 1 0 0 -1 -3],[ 1 1 1 0 1 0 1],[ 3 2 1 1 3 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,1,3,-1,0,0,1,2,0,1,1,1,0,0,1,1,3,-1]
Phi over symmetry [-3,-1,1,1,1,1,-1,1,1,2,3,0,1,1,1,0,0,0,-1,-1,0]
Phi of -K [-3,-1,1,1,1,1,3,1,2,3,3,1,1,1,2,0,-1,0,-1,0,0]
Phi of K* [-1,-1,-1,-1,1,3,-1,-1,0,1,3,0,0,1,1,0,1,2,2,3,3]
Phi of -K* [-3,-1,1,1,1,1,-1,1,1,2,3,0,1,1,1,0,0,0,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial -4w^4z^2+11w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+21t^4+20t^2
Outer characteristic polynomial t^7+35t^5+52t^3+8t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -3136*K1**4 + 1920*K1**3*K2*K3 + 128*K1**3*K3*K4 - 896*K1**3*K3 + 640*K1**2*K2**2*K4 - 6352*K1**2*K2**2 - 1184*K1**2*K2*K4 + 6912*K1**2*K2 - 1792*K1**2*K3**2 - 192*K1**2*K3*K5 - 416*K1**2*K4**2 - 1868*K1**2 + 1952*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 1248*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 800*K1*K2**2*K5 - 672*K1*K2*K3*K4 - 224*K1*K2*K3*K6 + 6560*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 1984*K1*K3*K4 + 456*K1*K4*K5 + 24*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4*K4**2 + 736*K2**4*K4 - 2304*K2**4 + 64*K2**3*K4*K6 - 256*K2**3*K6 - 1472*K2**2*K3**2 - 776*K2**2*K4**2 + 2016*K2**2*K4 - 16*K2**2*K5**2 - 48*K2**2*K6**2 - 1566*K2**2 + 984*K2*K3*K5 + 448*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 1464*K3**2 - 648*K4**2 - 160*K5**2 - 58*K6**2 - 4*K7**2 - 2*K8**2 + 2408
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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