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Flat knot 6.1290

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,1,4,0,0,0,1,0,1,1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1290']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 10*K1*K2 + 2*K1 + 5*K2 + 4*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1290']
Outer characteristic polynomial of the knot is: t^7+42t^5+36t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1290']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 320*K1**4*K2**2 + 2784*K1**4*K2 - 6960*K1**4 + 1984*K1**3*K2*K3 + 96*K1**3*K3*K4 - 2016*K1**3*K3 - 192*K1**2*K2**4 + 704*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 7680*K1**2*K2**2 - 1440*K1**2*K2*K4 + 12952*K1**2*K2 - 1904*K1**2*K3**2 - 64*K1**2*K3*K5 - 352*K1**2*K4**2 - 6032*K1**2 + 480*K1*K2**3*K3 - 1376*K1*K2**2*K3 - 480*K1*K2**2*K5 - 672*K1*K2*K3*K4 + 10064*K1*K2*K3 - 64*K1*K2*K4*K5 + 2896*K1*K3*K4 + 696*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 600*K2**4 - 32*K2**3*K6 - 384*K2**2*K3**2 - 168*K2**2*K4**2 + 1728*K2**2*K4 - 6112*K2**2 - 64*K2*K3**2*K4 + 856*K2*K3*K5 + 224*K2*K4*K6 + 48*K3**2*K6 - 3252*K3**2 - 1342*K4**2 - 404*K5**2 - 80*K6**2 + 6316
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1290']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4463', 'vk6.4560', 'vk6.5849', 'vk6.5978', 'vk6.7903', 'vk6.8023', 'vk6.9336', 'vk6.9457', 'vk6.13399', 'vk6.13494', 'vk6.13687', 'vk6.14077', 'vk6.15048', 'vk6.15170', 'vk6.17796', 'vk6.17829', 'vk6.18830', 'vk6.19414', 'vk6.19709', 'vk6.24343', 'vk6.25423', 'vk6.25456', 'vk6.26588', 'vk6.33253', 'vk6.33312', 'vk6.37557', 'vk6.44871', 'vk6.48652', 'vk6.50550', 'vk6.53653', 'vk6.55813', 'vk6.65485']
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 O1O2O3U2O4O5U1U5O6U4U6U3
R3 orbit {'O1O2O3U2O4O5U1U5O6U4U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U5O4U6U3O6O5U2
Gauss code of K* O1O2O3U4U5U3O5U1U6O4O6U2
Gauss code of -K* O1O2O3U2O4O5U4U3O6U1U6U5
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 2 0 1 1],[ 3 0 0 4 2 1 1],[ 1 0 0 1 0 0 0],[-2 -4 -1 0 -1 0 1],[ 0 -2 0 1 0 0 1],[-1 -1 0 0 0 0 0],[-1 -1 0 -1 -1 0 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 1 0 -1 -1 -4],[-1 -1 0 0 -1 0 -1],[-1 0 0 0 0 0 -1],[ 0 1 1 0 0 0 -2],[ 1 1 0 0 0 0 0],[ 3 4 1 1 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,-1,0,1,1,4,0,1,0,1,0,0,1,0,2,0]
Phi over symmetry [-3,-1,0,1,1,2,0,2,1,1,4,0,0,0,1,0,1,1,0,0,-1]
Phi of -K [-3,-1,0,1,1,2,2,1,3,3,1,1,2,2,2,0,1,1,0,2,1]
Phi of K* [-2,-1,-1,0,1,3,1,2,1,2,1,0,1,2,3,0,2,3,1,1,2]
Phi of -K* [-3,-1,0,1,1,2,0,2,1,1,4,0,0,0,1,0,1,1,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+26t^4+11t^2+1
Outer characteristic polynomial t^7+42t^5+36t^3+5t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 10*K1*K2 + 2*K1 + 5*K2 + 4*K3 + 6
2-strand cable arrow polynomial -384*K1**6 - 320*K1**4*K2**2 + 2784*K1**4*K2 - 6960*K1**4 + 1984*K1**3*K2*K3 + 96*K1**3*K3*K4 - 2016*K1**3*K3 - 192*K1**2*K2**4 + 704*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 7680*K1**2*K2**2 - 1440*K1**2*K2*K4 + 12952*K1**2*K2 - 1904*K1**2*K3**2 - 64*K1**2*K3*K5 - 352*K1**2*K4**2 - 6032*K1**2 + 480*K1*K2**3*K3 - 1376*K1*K2**2*K3 - 480*K1*K2**2*K5 - 672*K1*K2*K3*K4 + 10064*K1*K2*K3 - 64*K1*K2*K4*K5 + 2896*K1*K3*K4 + 696*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 600*K2**4 - 32*K2**3*K6 - 384*K2**2*K3**2 - 168*K2**2*K4**2 + 1728*K2**2*K4 - 6112*K2**2 - 64*K2*K3**2*K4 + 856*K2*K3*K5 + 224*K2*K4*K6 + 48*K3**2*K6 - 3252*K3**2 - 1342*K4**2 - 404*K5**2 - 80*K6**2 + 6316
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{4, 6}, {2, 5}, {1, 3}], [{6}, {2, 5}, {4}, {1, 3}]]
If K is slice False
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