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Flat knot 6.1166

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,1,2,2,3,1,1,1,1,0,1,0,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1166']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 6*K1*K2 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.552', '6.652', '6.764', '6.776', '6.784', '6.839', '6.903', '6.1010', '6.1166']
Outer characteristic polynomial of the knot is: t^7+59t^5+82t^3+14t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1166']
2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 1024*K1**4*K2 - 1152*K1**4 + 32*K1**3*K2*K3 - 64*K1**3*K3 - 384*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 2240*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 448*K1**2*K2**2*K4 - 8624*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 800*K1**2*K2*K4 + 7272*K1**2*K2 - 256*K1**2*K3**2 - 32*K1**2*K3*K5 - 4596*K1**2 + 1920*K1*K2**3*K3 + 768*K1*K2**2*K3*K4 - 2368*K1*K2**2*K3 - 672*K1*K2**2*K5 - 640*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8576*K1*K2*K3 + 1160*K1*K3*K4 + 72*K1*K4*K5 - 32*K2**6 + 416*K2**4*K4 - 3184*K2**4 - 64*K2**3*K6 - 2016*K2**2*K3**2 - 536*K2**2*K4**2 + 2976*K2**2*K4 - 2828*K2**2 + 1208*K2*K3*K5 + 104*K2*K4*K6 - 2132*K3**2 - 708*K4**2 - 120*K5**2 - 4*K6**2 + 3858
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1166']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16340', 'vk6.16381', 'vk6.18065', 'vk6.18403', 'vk6.22679', 'vk6.22756', 'vk6.24508', 'vk6.24931', 'vk6.34623', 'vk6.34704', 'vk6.36649', 'vk6.37073', 'vk6.42314', 'vk6.42343', 'vk6.43931', 'vk6.44250', 'vk6.54603', 'vk6.54640', 'vk6.55885', 'vk6.56173', 'vk6.59089', 'vk6.59126', 'vk6.60405', 'vk6.60764', 'vk6.64636', 'vk6.64680', 'vk6.65519', 'vk6.65835', 'vk6.67993', 'vk6.68017', 'vk6.68605', 'vk6.68822']
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 O1O2O3O4U1U5U2O5O6U3U4U6
R3 orbit {'O1O2O3O4U1U5U2O5O6U3U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1U2O5O6U3U6U4
Gauss code of K* O1O2O3U2U4O5O6O4U1U3U5U6
Gauss code of -K* O1O2O3U1U4O5O4O6U2U3U5U6
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 0 0 2 -1 2],[ 3 0 1 2 3 2 2],[ 0 -1 0 0 1 0 2],[ 0 -2 0 0 1 0 2],[-2 -3 -1 -1 0 -2 1],[ 1 -2 0 0 2 0 2],[-2 -2 -2 -2 -1 -2 0]]
Primitive based matrix [[ 0 2 2 0 0 -1 -3],[-2 0 1 -1 -1 -2 -3],[-2 -1 0 -2 -2 -2 -2],[ 0 1 2 0 0 0 -1],[ 0 1 2 0 0 0 -2],[ 1 2 2 0 0 0 -2],[ 3 3 2 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,1,3,-1,1,1,2,3,2,2,2,2,0,0,1,0,2,2]
Phi over symmetry [-3,-1,0,0,2,2,0,1,2,2,3,1,1,1,1,0,1,0,1,0,-1]
Phi of -K [-3,-1,0,0,2,2,0,1,2,2,3,1,1,1,1,0,1,0,1,0,-1]
Phi of K* [-2,-2,0,0,1,3,-1,0,0,1,3,1,1,1,2,0,1,1,1,2,0]
Phi of -K* [-3,-1,0,0,2,2,2,1,2,2,3,0,0,2,2,0,2,1,2,1,-1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+41t^4+34t^2+4
Outer characteristic polynomial t^7+59t^5+82t^3+14t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 6*K1*K2 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial -512*K1**4*K2**2 + 1024*K1**4*K2 - 1152*K1**4 + 32*K1**3*K2*K3 - 64*K1**3*K3 - 384*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 2240*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 448*K1**2*K2**2*K4 - 8624*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 800*K1**2*K2*K4 + 7272*K1**2*K2 - 256*K1**2*K3**2 - 32*K1**2*K3*K5 - 4596*K1**2 + 1920*K1*K2**3*K3 + 768*K1*K2**2*K3*K4 - 2368*K1*K2**2*K3 - 672*K1*K2**2*K5 - 640*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8576*K1*K2*K3 + 1160*K1*K3*K4 + 72*K1*K4*K5 - 32*K2**6 + 416*K2**4*K4 - 3184*K2**4 - 64*K2**3*K6 - 2016*K2**2*K3**2 - 536*K2**2*K4**2 + 2976*K2**2*K4 - 2828*K2**2 + 1208*K2*K3*K5 + 104*K2*K4*K6 - 2132*K3**2 - 708*K4**2 - 120*K5**2 - 4*K6**2 + 3858
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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