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Flat knot 6.920

Min(phi) over symmetries of the knot is: [-3,-1,1,3,1,1,2,2,3,1]
Flat knots (up to 7 crossings) with same phi are :['6.920']
Arrow polynomial of the knot is: 4*K1**3 + 8*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.521', '6.920', '6.1255', '6.1917']
Outer characteristic polynomial of the knot is: t^5+48t^3+44t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.920']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 256*K1**4*K2**3 - 768*K1**4*K2**2 + 896*K1**4*K2 - 1184*K1**4 + 320*K1**3*K2*K3 - 640*K1**2*K2**4 + 768*K1**2*K2**3 - 2928*K1**2*K2**2 + 2640*K1**2*K2 - 96*K1**2*K3**2 - 1544*K1**2 + 896*K1*K2**3*K3 + 2792*K1*K2*K3 + 224*K1*K3*K4 + 32*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 448*K2**4*K4 - 1168*K2**4 + 192*K2**3*K3*K5 + 128*K2**3*K4*K6 - 992*K2**2*K3**2 - 848*K2**2*K4**2 + 1136*K2**2*K4 - 112*K2**2*K5**2 - 16*K2**2*K6**2 - 1358*K2**2 + 616*K2*K3*K5 + 432*K2*K4*K6 + 32*K2*K5*K7 + 8*K3**2*K6 - 952*K3**2 - 500*K4**2 - 144*K5**2 - 58*K6**2 + 2026
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.920']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10919', 'vk6.10930', 'vk6.10934', 'vk6.12085', 'vk6.12089', 'vk6.12100', 'vk6.12104', 'vk6.14477', 'vk6.14478', 'vk6.15699', 'vk6.15700', 'vk6.16141', 'vk6.16142', 'vk6.30525', 'vk6.30529', 'vk6.30555', 'vk6.30559', 'vk6.31806', 'vk6.34070', 'vk6.34164', 'vk6.34165', 'vk6.34507', 'vk6.51763', 'vk6.51767', 'vk6.52636', 'vk6.54135', 'vk6.54136', 'vk6.54323', 'vk6.54527', 'vk6.63470', 'vk6.63481', 'vk6.63485']
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 O1O2O3O4U3U5O6O5U1U2U6U4
R3 orbit {'O1O2O3O4U3U5O6O5U1U2U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U3U4O6O5U6U2
Gauss code of K* O1O2O3O4U1U2U5U4O5O6U3U6
Gauss code of -K* O1O2O3O4U5U2O5O6U1U6U3U4
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 -1 -1 3 1 1],[ 3 0 1 0 4 3 1],[ 1 -1 0 0 3 1 0],[ 1 0 0 0 1 1 0],[-3 -4 -3 -1 0 -2 -1],[-1 -3 -1 -1 2 0 1],[-1 -1 0 0 1 -1 0]]
Primitive based matrix [[ 0 3 1 -1 -3],[-3 0 -1 -3 -4],[-1 1 0 0 -1],[ 1 3 0 0 -1],[ 3 4 1 1 0]]
If based matrix primitive False
Phi of primitive based matrix [-3,-1,1,3,1,3,4,0,1,1]
Phi over symmetry [-3,-1,1,3,1,1,2,2,3,1]
Phi of -K [-3,-1,1,3,1,3,2,2,1,1]
Phi of K* [-3,-1,1,3,1,1,2,2,3,1]
Phi of -K* [-3,-1,1,3,1,1,4,0,3,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 9z+19
Enhanced Jones-Krushkal polynomial -8w^3z+17w^2z+19w
Inner characteristic polynomial t^4+28t^2+4
Outer characteristic polynomial t^5+48t^3+44t
Flat arrow polynomial 4*K1**3 + 8*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -128*K1**6 + 256*K1**4*K2**3 - 768*K1**4*K2**2 + 896*K1**4*K2 - 1184*K1**4 + 320*K1**3*K2*K3 - 640*K1**2*K2**4 + 768*K1**2*K2**3 - 2928*K1**2*K2**2 + 2640*K1**2*K2 - 96*K1**2*K3**2 - 1544*K1**2 + 896*K1*K2**3*K3 + 2792*K1*K2*K3 + 224*K1*K3*K4 + 32*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 448*K2**4*K4 - 1168*K2**4 + 192*K2**3*K3*K5 + 128*K2**3*K4*K6 - 992*K2**2*K3**2 - 848*K2**2*K4**2 + 1136*K2**2*K4 - 112*K2**2*K5**2 - 16*K2**2*K6**2 - 1358*K2**2 + 616*K2*K3*K5 + 432*K2*K4*K6 + 32*K2*K5*K7 + 8*K3**2*K6 - 952*K3**2 - 500*K4**2 - 144*K5**2 - 58*K6**2 + 2026
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}]]
If K is slice False
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