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Flat knot 6.159

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,1,1,4,4,3,0,2,2,2,0,1,1,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.159']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 16*K1**2 - 6*K1*K2 - 2*K2**2 + 6*K2 + 2*K3 + 9
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.159', '6.321']
Outer characteristic polynomial of the knot is: t^7+89t^5+47t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.159']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 1024*K1**4*K2 - 4864*K1**4 + 768*K1**3*K2*K3 - 1344*K1**3*K3 - 128*K1**2*K2**4 + 704*K1**2*K2**3 + 416*K1**2*K2**2*K4 - 8336*K1**2*K2**2 + 160*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 992*K1**2*K2*K4 + 14120*K1**2*K2 - 1472*K1**2*K3**2 - 64*K1**2*K3*K5 - 320*K1**2*K4**2 - 8292*K1**2 + 768*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 2176*K1*K2**2*K3 - 288*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 736*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 11992*K1*K2*K3 + 2680*K1*K3*K4 + 416*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 288*K2**4*K4 - 1792*K2**4 + 32*K2**3*K3*K5 - 96*K2**3*K6 + 64*K2**2*K3**2*K4 - 1168*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 440*K2**2*K4**2 + 2800*K2**2*K4 - 6884*K2**2 + 888*K2*K3*K5 + 224*K2*K4*K6 - 96*K3**4 - 48*K3**2*K4**2 + 48*K3**2*K6 - 3592*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 1264*K4**2 - 188*K5**2 - 28*K6**2 + 7230
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.159']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16937', 'vk6.17178', 'vk6.20555', 'vk6.21956', 'vk6.23333', 'vk6.23626', 'vk6.28009', 'vk6.29476', 'vk6.35373', 'vk6.35792', 'vk6.39417', 'vk6.41610', 'vk6.42846', 'vk6.43123', 'vk6.45993', 'vk6.47669', 'vk6.55100', 'vk6.55355', 'vk6.57423', 'vk6.58594', 'vk6.59498', 'vk6.59792', 'vk6.62090', 'vk6.63068', 'vk6.64945', 'vk6.65151', 'vk6.66959', 'vk6.67820', 'vk6.68234', 'vk6.68375', 'vk6.69570', 'vk6.70267']
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 O1O2O3O4O5U1O6U3U6U5U2U4
R3 orbit {'O1O2O3O4O5U1O6U3U6U5U2U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U4U1U6U3O6U5
Gauss code of K* O1O2O3O4O5U6U4U1U5U3O6U2
Gauss code of -K* O1O2O3O4O5U4O6U3U1U5U2U6
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 -4 0 -2 3 2 1],[ 4 0 3 1 4 2 1],[ 0 -3 0 -2 2 1 1],[ 2 -1 2 0 3 2 1],[-3 -4 -2 -3 0 0 0],[-2 -2 -1 -2 0 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 3 2 1 0 -2 -4],[-3 0 0 0 -2 -3 -4],[-2 0 0 0 -1 -2 -2],[-1 0 0 0 -1 -1 -1],[ 0 2 1 1 0 -2 -3],[ 2 3 2 1 2 0 -1],[ 4 4 2 1 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,0,2,4,0,0,2,3,4,0,1,2,2,1,1,1,2,3,1]
Phi over symmetry [-4,-2,0,1,2,3,1,1,4,4,3,0,2,2,2,0,1,1,1,2,1]
Phi of -K [-4,-2,0,1,2,3,1,1,4,4,3,0,2,2,2,0,1,1,1,2,1]
Phi of K* [-3,-2,-1,0,2,4,1,2,1,2,3,1,1,2,4,0,2,4,0,1,1]
Phi of -K* [-4,-2,0,1,2,3,1,3,1,2,4,2,1,2,3,1,1,2,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+55t^4+15t^2+1
Outer characteristic polynomial t^7+89t^5+47t^3+5t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 16*K1**2 - 6*K1*K2 - 2*K2**2 + 6*K2 + 2*K3 + 9
2-strand cable arrow polynomial -64*K1**6 + 1024*K1**4*K2 - 4864*K1**4 + 768*K1**3*K2*K3 - 1344*K1**3*K3 - 128*K1**2*K2**4 + 704*K1**2*K2**3 + 416*K1**2*K2**2*K4 - 8336*K1**2*K2**2 + 160*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 992*K1**2*K2*K4 + 14120*K1**2*K2 - 1472*K1**2*K3**2 - 64*K1**2*K3*K5 - 320*K1**2*K4**2 - 8292*K1**2 + 768*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 2176*K1*K2**2*K3 - 288*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 736*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 11992*K1*K2*K3 + 2680*K1*K3*K4 + 416*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 288*K2**4*K4 - 1792*K2**4 + 32*K2**3*K3*K5 - 96*K2**3*K6 + 64*K2**2*K3**2*K4 - 1168*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 440*K2**2*K4**2 + 2800*K2**2*K4 - 6884*K2**2 + 888*K2*K3*K5 + 224*K2*K4*K6 - 96*K3**4 - 48*K3**2*K4**2 + 48*K3**2*K6 - 3592*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 1264*K4**2 - 188*K5**2 - 28*K6**2 + 7230
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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