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Flat knot 6.270

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,0,3,1,3,4,2,1,2,2,0,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.270']
Arrow polynomial of the knot is: -4*K1**2 - 6*K1*K2 + 3*K1 - 2*K2**2 + 2*K2 + 3*K3 + K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.147', '6.270']
Outer characteristic polynomial of the knot is: t^7+99t^5+64t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.270']
2-strand cable arrow polynomial of the knot is: 192*K1**4*K2 - 352*K1**4 + 64*K1**3*K2*K3 - 960*K1**3*K3 - 64*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 848*K1**2*K2**2 + 96*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 544*K1**2*K2*K4 + 4328*K1**2*K2 - 656*K1**2*K3**2 - 32*K1**2*K3*K5 - 80*K1**2*K4**2 - 5052*K1**2 + 224*K1*K2**3*K3 - 992*K1*K2**2*K3 - 64*K1*K2**2*K5 + 64*K1*K2*K3**3 - 352*K1*K2*K3*K4 + 4960*K1*K2*K3 - 96*K1*K2*K4*K5 + 64*K1*K3**3*K4 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2392*K1*K3*K4 + 280*K1*K4*K5 + 32*K1*K5*K6 - 192*K2**4 - 544*K2**2*K3**2 - 56*K2**2*K4**2 + 1064*K2**2*K4 - 3514*K2**2 - 64*K2*K3**2*K4 + 696*K2*K3*K5 + 144*K2*K4*K6 - 112*K3**4 - 64*K3**2*K4**2 + 128*K3**2*K6 - 2456*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1204*K4**2 - 260*K5**2 - 70*K6**2 - 8*K7**2 - 2*K8**2 + 3828
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.270']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73333', 'vk6.73334', 'vk6.73495', 'vk6.73497', 'vk6.75256', 'vk6.75257', 'vk6.75503', 'vk6.75505', 'vk6.78220', 'vk6.78222', 'vk6.78461', 'vk6.78462', 'vk6.80045', 'vk6.80047', 'vk6.80195', 'vk6.80196', 'vk6.81929', 'vk6.81940', 'vk6.82193', 'vk6.82209', 'vk6.82657', 'vk6.82664', 'vk6.84719', 'vk6.84726', 'vk6.85023', 'vk6.85026', 'vk6.85747', 'vk6.86503', 'vk6.87322', 'vk6.87679', 'vk6.89629', 'vk6.90084']
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 O1O2O3O4O5U1U4O6U2U5U3U6
R3 orbit {'O1O2O3O4O5U1U4O6U2U5U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U3U1U4O6U2U5
Gauss code of K* O1O2O3O4U5U1U3U6U2O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U3U5U2U4U6
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 1 0 2 3],[ 4 0 2 4 1 3 3],[ 2 -2 0 2 0 2 3],[-1 -4 -2 0 -1 1 2],[ 0 -1 0 1 0 1 1],[-2 -3 -2 -1 -1 0 1],[-3 -3 -3 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 2 1 0 -2 -4],[-3 0 -1 -2 -1 -3 -3],[-2 1 0 -1 -1 -2 -3],[-1 2 1 0 -1 -2 -4],[ 0 1 1 1 0 0 -1],[ 2 3 2 2 0 0 -2],[ 4 3 3 4 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,0,2,4,1,2,1,3,3,1,1,2,3,1,2,4,0,1,2]
Phi over symmetry [-4,-2,0,1,2,3,0,3,1,3,4,2,1,2,2,0,1,2,0,0,0]
Phi of -K [-4,-2,0,1,2,3,0,3,1,3,4,2,1,2,2,0,1,2,0,0,0]
Phi of K* [-3,-2,-1,0,2,4,0,0,2,2,4,0,1,2,3,0,1,1,2,3,0]
Phi of -K* [-4,-2,0,1,2,3,2,1,4,3,3,0,2,2,3,1,1,1,1,2,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 3z^2+20z+29
Enhanced Jones-Krushkal polynomial 3w^3z^2+20w^2z+29w
Inner characteristic polynomial t^6+65t^4+12t^2
Outer characteristic polynomial t^7+99t^5+64t^3+4t
Flat arrow polynomial -4*K1**2 - 6*K1*K2 + 3*K1 - 2*K2**2 + 2*K2 + 3*K3 + K4 + 4
2-strand cable arrow polynomial 192*K1**4*K2 - 352*K1**4 + 64*K1**3*K2*K3 - 960*K1**3*K3 - 64*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 848*K1**2*K2**2 + 96*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 544*K1**2*K2*K4 + 4328*K1**2*K2 - 656*K1**2*K3**2 - 32*K1**2*K3*K5 - 80*K1**2*K4**2 - 5052*K1**2 + 224*K1*K2**3*K3 - 992*K1*K2**2*K3 - 64*K1*K2**2*K5 + 64*K1*K2*K3**3 - 352*K1*K2*K3*K4 + 4960*K1*K2*K3 - 96*K1*K2*K4*K5 + 64*K1*K3**3*K4 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2392*K1*K3*K4 + 280*K1*K4*K5 + 32*K1*K5*K6 - 192*K2**4 - 544*K2**2*K3**2 - 56*K2**2*K4**2 + 1064*K2**2*K4 - 3514*K2**2 - 64*K2*K3**2*K4 + 696*K2*K3*K5 + 144*K2*K4*K6 - 112*K3**4 - 64*K3**2*K4**2 + 128*K3**2*K6 - 2456*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1204*K4**2 - 260*K5**2 - 70*K6**2 - 8*K7**2 - 2*K8**2 + 3828
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}]]
If K is slice False
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