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Flat knot 6.1254

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,1,2,0,0,1,2,1,2,2,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.1254']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1*K3 + K4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.111', '6.139', '6.519', '6.566', '6.1228', '6.1254', '6.1259', '6.1912', '6.1936']
Outer characteristic polynomial of the knot is: t^7+38t^5+113t^3+16t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1254']
2-strand cable arrow polynomial of the knot is: -1600*K1**2*K2**2 - 192*K1**2*K2*K4 + 2128*K1**2*K2 - 512*K1**2*K3**2 - 2656*K1**2 + 704*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 832*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 256*K1*K2*K3*K6 + 4976*K1*K2*K3 + 896*K1*K3*K4 + 16*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 976*K2**4 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 2176*K2**2*K3**2 - 256*K2**2*K4**2 + 1184*K2**2*K4 - 48*K2**2*K6**2 - 2384*K2**2 + 1824*K2*K3*K5 + 224*K2*K4*K6 + 16*K2*K6*K8 + 48*K3**2*K6 - 2000*K3**2 - 440*K4**2 - 368*K5**2 - 48*K6**2 - 2*K8**2 + 2632
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1254']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73623', 'vk6.73637', 'vk6.74411', 'vk6.75025', 'vk6.75580', 'vk6.75605', 'vk6.76596', 'vk6.76957', 'vk6.78549', 'vk6.78575', 'vk6.79447', 'vk6.79861', 'vk6.80239', 'vk6.80890', 'vk6.83708', 'vk6.84855', 'vk6.85674', 'vk6.87620', 'vk6.88438', 'vk6.89290']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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 O1O2O3O4U5U2U1O5O6U4U3U6
R3 orbit {'O1O2O3O4U5U2U1O5O6U4U3U6'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2O3U1U4O5O6O4U3U2U6U5
Gauss code of -K* O1O2O3U1U4O5O6O4U3U2U6U5
Diagrammatic symmetry type r
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 -1 1 1 -2 2],[ 1 0 0 2 1 0 2],[ 1 0 0 1 0 1 2],[-1 -2 -1 0 0 -1 2],[-1 -1 0 0 0 -1 1],[ 2 0 -1 1 1 0 2],[-2 -2 -2 -2 -1 -2 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 -1 -2 -2 -2 -2],[-1 1 0 0 0 -1 -1],[-1 2 0 0 -1 -2 -1],[ 1 2 0 1 0 0 1],[ 1 2 1 2 0 0 0],[ 2 2 1 1 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,1,2,2,2,2,0,0,1,1,1,2,1,0,-1,0]
Phi over symmetry [-2,-1,-1,1,1,2,-1,0,1,1,2,0,0,1,2,1,2,2,0,1,2]
Phi of -K [-2,-1,-1,1,1,2,1,2,2,2,2,0,0,1,1,1,2,1,0,-1,0]
Phi of K* [-2,-1,-1,1,1,2,-1,0,1,1,2,0,0,1,2,1,2,2,0,1,2]
Phi of -K* [-2,-1,-1,1,1,2,-1,0,1,1,2,0,0,1,2,1,2,2,0,1,2]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+25z+19
Enhanced Jones-Krushkal polynomial -2w^4z^2+10w^3z^2-4w^3z+29w^2z+19w
Inner characteristic polynomial t^6+26t^4+39t^2+4
Outer characteristic polynomial t^7+38t^5+113t^3+16t
Flat arrow polynomial 4*K1**2*K2 - 4*K1*K3 + K4
2-strand cable arrow polynomial -1600*K1**2*K2**2 - 192*K1**2*K2*K4 + 2128*K1**2*K2 - 512*K1**2*K3**2 - 2656*K1**2 + 704*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 832*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 256*K1*K2*K3*K6 + 4976*K1*K2*K3 + 896*K1*K3*K4 + 16*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 976*K2**4 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 2176*K2**2*K3**2 - 256*K2**2*K4**2 + 1184*K2**2*K4 - 48*K2**2*K6**2 - 2384*K2**2 + 1824*K2*K3*K5 + 224*K2*K4*K6 + 16*K2*K6*K8 + 48*K3**2*K6 - 2000*K3**2 - 440*K4**2 - 368*K5**2 - 48*K6**2 - 2*K8**2 + 2632
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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