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Flat knot 6.1226

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,2,2,2,2,-1,0,1,0,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1226']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1030', '6.1062', '6.1226', '6.1508', '6.1525', '6.1596', '6.1724', '6.1729', '6.1735', '6.1738', '6.1789', '6.1809', '6.1921']
Outer characteristic polynomial of the knot is: t^7+34t^5+58t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1226']
2-strand cable arrow polynomial of the knot is: -288*K1**4 - 128*K1**3*K3 + 480*K1**2*K2**3 - 1904*K1**2*K2**2 - 160*K1**2*K2*K4 + 2480*K1**2*K2 - 256*K1**2*K3**2 - 32*K1**2*K4**2 - 2204*K1**2 + 512*K1*K2**3*K3 - 960*K1*K2**2*K3 - 96*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 3136*K1*K2*K3 + 1040*K1*K3*K4 + 160*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 952*K2**4 - 32*K2**3*K6 - 608*K2**2*K3**2 - 128*K2**2*K4**2 + 1416*K2**2*K4 - 1834*K2**2 + 608*K2*K3*K5 + 104*K2*K4*K6 - 1324*K3**2 - 746*K4**2 - 200*K5**2 - 22*K6**2 + 2128
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1226']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20126', 'vk6.20130', 'vk6.21414', 'vk6.21422', 'vk6.27224', 'vk6.27232', 'vk6.28894', 'vk6.28898', 'vk6.38633', 'vk6.38641', 'vk6.40839', 'vk6.40853', 'vk6.45526', 'vk6.45542', 'vk6.47240', 'vk6.47247', 'vk6.56947', 'vk6.56951', 'vk6.58099', 'vk6.58107', 'vk6.61515', 'vk6.61523', 'vk6.62642', 'vk6.62646', 'vk6.66649', 'vk6.66651', 'vk6.67442', 'vk6.67446', 'vk6.69293', 'vk6.69297', 'vk6.70022', 'vk6.70024']
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 O1O2O3O4U3U2U1O5O6U4U5U6
R3 orbit {'O1O2O3O4U3U2U1O5O6U4U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U1O5O6U4U3U2
Gauss code of K* O1O2O3U4U5O6O4O5U3U2U1U6
Gauss code of -K* O1O2O3U1U2O4O5O6U3U6U5U4
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 -1 -1 -1 1 0 2],[ 1 0 0 0 3 1 1],[ 1 0 0 0 2 1 1],[ 1 0 0 0 1 1 1],[-1 -3 -2 -1 0 1 2],[ 0 -1 -1 -1 -1 0 1],[-2 -1 -1 -1 -2 -1 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 -2 -1 -1 -1 -1],[-1 2 0 1 -1 -2 -3],[ 0 1 -1 0 -1 -1 -1],[ 1 1 1 1 0 0 0],[ 1 1 2 1 0 0 0],[ 1 1 3 1 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,2,1,1,1,1,-1,1,2,3,1,1,1,0,0,0]
Phi over symmetry [-2,-1,0,1,1,1,-1,1,2,2,2,2,-1,0,1,0,0,0,0,0,0]
Phi of -K [-1,-1,-1,0,1,2,0,0,0,-1,2,0,0,0,2,0,1,2,2,1,-1]
Phi of K* [-2,-1,0,1,1,1,-1,1,2,2,2,2,-1,0,1,0,0,0,0,0,0]
Phi of -K* [-1,-1,-1,0,1,2,0,0,1,1,1,0,1,2,1,1,3,1,-1,1,2]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial z^2+6z+9
Enhanced Jones-Krushkal polynomial -4w^4z^2+5w^3z^2-12w^3z+18w^2z+9w
Inner characteristic polynomial t^6+26t^4+17t^2
Outer characteristic polynomial t^7+34t^5+58t^3+6t
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2
2-strand cable arrow polynomial -288*K1**4 - 128*K1**3*K3 + 480*K1**2*K2**3 - 1904*K1**2*K2**2 - 160*K1**2*K2*K4 + 2480*K1**2*K2 - 256*K1**2*K3**2 - 32*K1**2*K4**2 - 2204*K1**2 + 512*K1*K2**3*K3 - 960*K1*K2**2*K3 - 96*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 3136*K1*K2*K3 + 1040*K1*K3*K4 + 160*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 952*K2**4 - 32*K2**3*K6 - 608*K2**2*K3**2 - 128*K2**2*K4**2 + 1416*K2**2*K4 - 1834*K2**2 + 608*K2*K3*K5 + 104*K2*K4*K6 - 1324*K3**2 - 746*K4**2 - 200*K5**2 - 22*K6**2 + 2128
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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