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Flat knot 6.218

Min(phi) over symmetries of the knot is: [-3,-1,0,0,1,3,-1,1,1,3,3,1,1,1,1,-1,0,1,1,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.218']
Arrow polynomial of the knot is: 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.218', '6.554', '6.929', '6.932', '6.1014', '6.1024', '6.1068', '6.1526', '6.1664', '6.1676', '6.1755', '6.1763', '6.2065', '6.2078']
Outer characteristic polynomial of the knot is: t^7+56t^5+84t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.218']
2-strand cable arrow polynomial of the knot is: -192*K1**6 - 256*K1**4*K2**2 + 1440*K1**4*K2 - 4032*K1**4 + 416*K1**3*K2*K3 - 672*K1**3*K3 - 320*K1**2*K2**4 + 1088*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 96*K1**2*K2**2*K4 - 8176*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 11776*K1**2*K2 - 1056*K1**2*K3**2 - 64*K1**2*K4**2 - 6372*K1**2 + 608*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1856*K1*K2**2*K3 - 128*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9672*K1*K2*K3 + 1776*K1*K3*K4 + 112*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 1584*K2**4 - 1104*K2**2*K3**2 - 112*K2**2*K4**2 + 1992*K2**2*K4 - 5120*K2**2 + 784*K2*K3*K5 + 40*K2*K4*K6 - 96*K3**4 + 64*K3**2*K6 - 2572*K3**2 - 772*K4**2 - 112*K5**2 - 16*K6**2 + 5538
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.218']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4204', 'vk6.4284', 'vk6.5460', 'vk6.5571', 'vk6.7565', 'vk6.7652', 'vk6.9066', 'vk6.9146', 'vk6.11175', 'vk6.12259', 'vk6.12366', 'vk6.19386', 'vk6.19681', 'vk6.19788', 'vk6.26166', 'vk6.26223', 'vk6.26584', 'vk6.26666', 'vk6.30765', 'vk6.31966', 'vk6.38170', 'vk6.38203', 'vk6.44827', 'vk6.44944', 'vk6.48526', 'vk6.49222', 'vk6.49329', 'vk6.50310', 'vk6.52745', 'vk6.63581', 'vk6.66322', 'vk6.66343']
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 O1O2O3O4O5U2O6U5U6U3U1U4
R3 orbit {'O1O2O3O4O5U2O6U5U6U3U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U5U3U6U1O6U4
Gauss code of K* O1O2O3O4O5U4U6U3U5U1O6U2
Gauss code of -K* O1O2O3O4O5U4O6U5U1U3U6U2
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 -3 0 3 0 1],[ 1 0 -2 1 3 0 1],[ 3 2 0 2 3 1 1],[ 0 -1 -2 0 1 -1 1],[-3 -3 -3 -1 0 -1 1],[ 0 0 -1 1 1 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 1 0 0 -1 -3],[-3 0 1 -1 -1 -3 -3],[-1 -1 0 -1 -1 -1 -1],[ 0 1 1 0 1 0 -1],[ 0 1 1 -1 0 -1 -2],[ 1 3 1 0 1 0 -2],[ 3 3 1 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,1,3,-1,1,1,3,3,1,1,1,1,-1,0,1,1,2,2]
Phi over symmetry [-3,-1,0,0,1,3,-1,1,1,3,3,1,1,1,1,-1,0,1,1,2,2]
Phi of -K [-3,-1,0,0,1,3,0,1,2,3,3,0,1,1,1,1,0,2,0,2,3]
Phi of K* [-3,-1,0,0,1,3,3,2,2,1,3,0,0,1,3,-1,0,1,1,2,0]
Phi of -K* [-3,-1,0,0,1,3,2,1,2,1,3,0,1,1,3,1,1,1,1,1,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 3z^2+23z+35
Enhanced Jones-Krushkal polynomial 3w^3z^2+23w^2z+35w
Inner characteristic polynomial t^6+36t^4+24t^2+1
Outer characteristic polynomial t^7+56t^5+84t^3+7t
Flat arrow polynomial 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -192*K1**6 - 256*K1**4*K2**2 + 1440*K1**4*K2 - 4032*K1**4 + 416*K1**3*K2*K3 - 672*K1**3*K3 - 320*K1**2*K2**4 + 1088*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 96*K1**2*K2**2*K4 - 8176*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 11776*K1**2*K2 - 1056*K1**2*K3**2 - 64*K1**2*K4**2 - 6372*K1**2 + 608*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1856*K1*K2**2*K3 - 128*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9672*K1*K2*K3 + 1776*K1*K3*K4 + 112*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 1584*K2**4 - 1104*K2**2*K3**2 - 112*K2**2*K4**2 + 1992*K2**2*K4 - 5120*K2**2 + 784*K2*K3*K5 + 40*K2*K4*K6 - 96*K3**4 + 64*K3**2*K6 - 2572*K3**2 - 772*K4**2 - 112*K5**2 - 16*K6**2 + 5538
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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