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Flat knot 6.1231

Min(phi) over symmetries of the knot is: [-1,-1,-1,1,1,1,-1,-1,1,1,1,0,1,1,1,1,2,2,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1231']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1*K2 + K1 + 3*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1231', '6.1372', '6.1722', '6.1817', '6.1862', '6.2082']
Outer characteristic polynomial of the knot is: t^7+25t^5+29t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1231']
2-strand cable arrow polynomial of the knot is: 1248*K1**4*K2 - 2592*K1**4 + 672*K1**3*K2*K3 - 1120*K1**3*K3 - 128*K1**2*K2**4 + 352*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3552*K1**2*K2**2 - 640*K1**2*K2*K4 + 4720*K1**2*K2 - 992*K1**2*K3**2 - 32*K1**2*K4**2 - 2224*K1**2 + 352*K1*K2**3*K3 - 384*K1*K2**2*K3 - 192*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 4528*K1*K2*K3 + 1352*K1*K3*K4 + 136*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 464*K2**4 - 32*K2**3*K6 - 256*K2**2*K3**2 - 128*K2**2*K4**2 + 832*K2**2*K4 - 2122*K2**2 + 384*K2*K3*K5 + 104*K2*K4*K6 - 1400*K3**2 - 608*K4**2 - 136*K5**2 - 22*K6**2 + 2374
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1231']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10529', 'vk6.10533', 'vk6.10618', 'vk6.10626', 'vk6.10805', 'vk6.10813', 'vk6.10904', 'vk6.10908', 'vk6.19020', 'vk6.19036', 'vk6.19088', 'vk6.19090', 'vk6.19135', 'vk6.19137', 'vk6.25542', 'vk6.25558', 'vk6.25637', 'vk6.25653', 'vk6.25760', 'vk6.25762', 'vk6.30214', 'vk6.30218', 'vk6.30305', 'vk6.30313', 'vk6.30432', 'vk6.30440', 'vk6.37722', 'vk6.37738', 'vk6.56509', 'vk6.56517', 'vk6.66165', 'vk6.66173']
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 O1O2O3O4U3U2U5O6O5U4U1U6
R3 orbit {'O1O2O3O4U3U2U5O6O5U4U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4U1O6O5U6U3U2
Gauss code of K* O1O2O3U4U3O5O6O4U6U2U1U5
Gauss code of -K* O1O2O3U4U1O4O5O6U3U6U5U2
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 1 1],[ 1 0 -1 -1 2 1 1],[ 1 1 0 0 2 1 1],[ 1 1 0 0 1 1 1],[-1 -2 -2 -1 0 -1 0],[-1 -1 -1 -1 1 0 1],[-1 -1 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 -1 -1 -1],[-1 0 1 1 -1 -1 -1],[-1 -1 0 0 -1 -1 -1],[-1 -1 0 0 -1 -2 -2],[ 1 1 1 1 0 1 0],[ 1 1 1 2 -1 0 -1],[ 1 1 1 2 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,1,1,1,-1,-1,1,1,1,0,1,1,1,1,2,2,-1,0,1]
Phi over symmetry [-1,-1,-1,1,1,1,-1,-1,1,1,1,0,1,1,1,1,2,2,-1,0,1]
Phi of -K [-1,-1,-1,1,1,1,-1,0,0,1,1,1,0,1,1,1,1,1,0,1,1]
Phi of K* [-1,-1,-1,1,1,1,-1,0,0,0,1,1,1,1,1,1,1,1,-1,-1,0]
Phi of -K* [-1,-1,-1,1,1,1,-1,-1,1,1,2,0,1,1,1,1,1,2,-1,0,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial 6w^3z^2+23w^2z+23w
Inner characteristic polynomial t^6+19t^4+9t^2+1
Outer characteristic polynomial t^7+25t^5+29t^3+5t
Flat arrow polynomial 4*K1**3 - 8*K1*K2 + K1 + 3*K3 + 1
2-strand cable arrow polynomial 1248*K1**4*K2 - 2592*K1**4 + 672*K1**3*K2*K3 - 1120*K1**3*K3 - 128*K1**2*K2**4 + 352*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3552*K1**2*K2**2 - 640*K1**2*K2*K4 + 4720*K1**2*K2 - 992*K1**2*K3**2 - 32*K1**2*K4**2 - 2224*K1**2 + 352*K1*K2**3*K3 - 384*K1*K2**2*K3 - 192*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 4528*K1*K2*K3 + 1352*K1*K3*K4 + 136*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 464*K2**4 - 32*K2**3*K6 - 256*K2**2*K3**2 - 128*K2**2*K4**2 + 832*K2**2*K4 - 2122*K2**2 + 384*K2*K3*K5 + 104*K2*K4*K6 - 1400*K3**2 - 608*K4**2 - 136*K5**2 - 22*K6**2 + 2374
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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