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Flat knot 6.1510

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,1,2,1,1,0,0,0,1,1,1,1,1,3,0]
Flat knots (up to 7 crossings) with same phi are :['6.1510']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 4*K1*K2 - K1 + 6*K2 + K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1345', '6.1510', '6.1565', '6.1691', '6.1812']
Outer characteristic polynomial of the knot is: t^7+31t^5+59t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1510']
2-strand cable arrow polynomial of the knot is: -1152*K1**4*K2**2 + 2400*K1**4*K2 - 3168*K1**4 + 320*K1**3*K2*K3 - 192*K1**3*K3 - 576*K1**2*K2**4 + 3648*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 11520*K1**2*K2**2 - 512*K1**2*K2*K4 + 10656*K1**2*K2 - 160*K1**2*K3**2 - 5476*K1**2 + 576*K1*K2**3*K3 - 2016*K1*K2**2*K3 - 192*K1*K2**2*K5 + 8512*K1*K2*K3 + 616*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 2752*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 16*K2**2*K4**2 + 2264*K2**2*K4 - 3638*K2**2 + 240*K2*K3*K5 + 16*K2*K4*K6 - 1796*K3**2 - 500*K4**2 - 64*K5**2 - 2*K6**2 + 4626
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1510']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13368', 'vk6.13435', 'vk6.13626', 'vk6.13750', 'vk6.14150', 'vk6.14377', 'vk6.15608', 'vk6.16076', 'vk6.16471', 'vk6.16488', 'vk6.17641', 'vk6.22882', 'vk6.22915', 'vk6.24190', 'vk6.33119', 'vk6.33152', 'vk6.33216', 'vk6.33279', 'vk6.34863', 'vk6.34896', 'vk6.36445', 'vk6.42445', 'vk6.42461', 'vk6.43543', 'vk6.53552', 'vk6.53591', 'vk6.53624', 'vk6.53686', 'vk6.54720', 'vk6.55687', 'vk6.60237', 'vk6.64575']
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 O1O2O3U2O4U1U4O5O6U3U5U6
R3 orbit {'O1O2O3U2O4U1U4O5O6U3U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5U1O4O5U6U3O6U2
Gauss code of K* O1O2O3U4U5U1O5U6O4O6U2U3
Gauss code of -K* O1O2O3U1U2O4O5U4O6U3U6U5
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 -2 -1 0 1 0 2],[ 2 0 0 3 1 1 1],[ 1 0 0 1 0 1 1],[ 0 -3 -1 0 0 1 2],[-1 -1 0 0 0 0 0],[ 0 -1 -1 -1 0 0 1],[-2 -1 -1 -2 0 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -2 -1 -1],[-1 0 0 0 0 0 -1],[ 0 1 0 0 -1 -1 -1],[ 0 2 0 1 0 -1 -3],[ 1 1 0 1 1 0 0],[ 2 1 1 1 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,1,2,1,1,0,0,0,1,1,1,1,1,3,0]
Phi over symmetry [-2,-1,0,0,1,2,0,1,2,1,1,0,0,0,1,1,1,1,1,3,0]
Phi of -K [-2,-1,0,0,1,2,1,-1,1,2,3,0,0,2,2,-1,1,0,1,1,1]
Phi of K* [-2,-1,0,0,1,2,1,0,1,2,3,1,1,2,2,1,0,-1,0,1,1]
Phi of -K* [-2,-1,0,0,1,2,0,1,3,1,1,1,1,0,1,-1,0,1,0,2,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+21t^4+21t^2
Outer characteristic polynomial t^7+31t^5+59t^3+10t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 4*K1*K2 - K1 + 6*K2 + K3 + 7
2-strand cable arrow polynomial -1152*K1**4*K2**2 + 2400*K1**4*K2 - 3168*K1**4 + 320*K1**3*K2*K3 - 192*K1**3*K3 - 576*K1**2*K2**4 + 3648*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 11520*K1**2*K2**2 - 512*K1**2*K2*K4 + 10656*K1**2*K2 - 160*K1**2*K3**2 - 5476*K1**2 + 576*K1*K2**3*K3 - 2016*K1*K2**2*K3 - 192*K1*K2**2*K5 + 8512*K1*K2*K3 + 616*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 2752*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 16*K2**2*K4**2 + 2264*K2**2*K4 - 3638*K2**2 + 240*K2*K3*K5 + 16*K2*K4*K6 - 1796*K3**2 - 500*K4**2 - 64*K5**2 - 2*K6**2 + 4626
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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