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Flat knot 6.256

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,1,1,3,2,4,0,2,1,2,1,1,0,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.256']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 4*K1**2*K2 - 2*K1*K2 - 2*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.152', '6.164', '6.256', '6.433']
Outer characteristic polynomial of the knot is: t^7+111t^5+118t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.256']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2 - 2016*K1**4 + 256*K1**3*K2*K3 - 416*K1**3*K3 + 128*K1**2*K2**5 - 1152*K1**2*K2**4 + 2208*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 8784*K1**2*K2**2 - 672*K1**2*K2*K4 + 8704*K1**2*K2 - 224*K1**2*K3**2 - 4096*K1**2 + 128*K1*K2**5*K3 + 1984*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 - 416*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 6880*K1*K2*K3 + 376*K1*K3*K4 + 24*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 1568*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 1536*K2**4*K4 - 3904*K2**4 + 32*K2**3*K3*K5 - 192*K2**3*K6 - 848*K2**2*K3**2 - 296*K2**2*K4**2 + 2800*K2**2*K4 - 1304*K2**2 + 336*K2*K3*K5 + 32*K2*K4*K6 - 1236*K3**2 - 312*K4**2 - 12*K5**2 + 3238
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.256']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81912', 'vk6.81913', 'vk6.81962', 'vk6.81965', 'vk6.82112', 'vk6.82115', 'vk6.82129', 'vk6.82134', 'vk6.82636', 'vk6.82641', 'vk6.82693', 'vk6.82696', 'vk6.82793', 'vk6.82799', 'vk6.83072', 'vk6.83076', 'vk6.83080', 'vk6.83536', 'vk6.84666', 'vk6.84667', 'vk6.84988', 'vk6.84993', 'vk6.85838', 'vk6.85841', 'vk6.86214', 'vk6.88475', 'vk6.88478', 'vk6.89082', 'vk6.89092', 'vk6.89095', 'vk6.89646', 'vk6.90059']
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 O1O2O3O4O5U1U2O6U5U3U4U6
R3 orbit {'O1O2O3O4O5U1U2O6U5U3U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U2U3U1O6U4U5
Gauss code of K* O1O2O3O4U5U6U2U3U1O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U4U2U3U5U6
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 -2 0 2 1 3],[ 4 0 1 3 4 2 3],[ 2 -1 0 2 3 1 3],[ 0 -3 -2 0 1 0 3],[-2 -4 -3 -1 0 0 2],[-1 -2 -1 0 0 0 1],[-3 -3 -3 -3 -2 -1 0]]
Primitive based matrix [[ 0 3 2 1 0 -2 -4],[-3 0 -2 -1 -3 -3 -3],[-2 2 0 0 -1 -3 -4],[-1 1 0 0 0 -1 -2],[ 0 3 1 0 0 -2 -3],[ 2 3 3 1 2 0 -1],[ 4 3 4 2 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,0,2,4,2,1,3,3,3,0,1,3,4,0,1,2,2,3,1]
Phi over symmetry [-4,-2,0,1,2,3,1,1,3,2,4,0,2,1,2,1,1,0,1,1,-1]
Phi of -K [-4,-2,0,1,2,3,1,1,3,2,4,0,2,1,2,1,1,0,1,1,-1]
Phi of K* [-3,-2,-1,0,2,4,-1,1,0,2,4,1,1,1,2,1,2,3,0,1,1]
Phi of -K* [-4,-2,0,1,2,3,1,3,2,4,3,2,1,3,3,0,1,3,0,1,2]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial -4w^4z^2+10w^3z^2+23w^2z+23w
Inner characteristic polynomial t^6+77t^4+40t^2+1
Outer characteristic polynomial t^7+111t^5+118t^3+13t
Flat arrow polynomial -8*K1**4 + 4*K1**3 + 4*K1**2*K2 - 2*K1*K2 - 2*K1 + 2*K2 + 3
2-strand cable arrow polynomial 256*K1**4*K2 - 2016*K1**4 + 256*K1**3*K2*K3 - 416*K1**3*K3 + 128*K1**2*K2**5 - 1152*K1**2*K2**4 + 2208*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 8784*K1**2*K2**2 - 672*K1**2*K2*K4 + 8704*K1**2*K2 - 224*K1**2*K3**2 - 4096*K1**2 + 128*K1*K2**5*K3 + 1984*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 - 416*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 6880*K1*K2*K3 + 376*K1*K3*K4 + 24*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 1568*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 1536*K2**4*K4 - 3904*K2**4 + 32*K2**3*K3*K5 - 192*K2**3*K6 - 848*K2**2*K3**2 - 296*K2**2*K4**2 + 2800*K2**2*K4 - 1304*K2**2 + 336*K2*K3*K5 + 32*K2*K4*K6 - 1236*K3**2 - 312*K4**2 - 12*K5**2 + 3238
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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