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Flat knot 6.143

Min(phi) over symmetries of the knot is: [-4,-3,-1,2,3,3,0,1,3,3,4,1,3,2,3,2,1,2,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.143']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.143', '6.158', '6.264', '6.282', '6.501']
Outer characteristic polynomial of the knot is: t^7+134t^5+148t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.143']
2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 1024*K1**4*K2 - 2528*K1**4 + 128*K1**3*K2**3*K3 - 256*K1**3*K2**2*K3 + 544*K1**3*K2*K3 - 288*K1**3*K3 - 384*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2496*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 9648*K1**2*K2**2 + 160*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 864*K1**2*K2*K4 + 10288*K1**2*K2 - 608*K1**2*K3**2 - 48*K1**2*K4**2 - 6428*K1**2 - 128*K1*K2**4*K3 + 2240*K1*K2**3*K3 + 672*K1*K2**2*K3*K4 - 1760*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 640*K1*K2**2*K5 + 64*K1*K2*K3**3 - 320*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8992*K1*K2*K3 - 32*K1*K2*K4*K5 + 1176*K1*K3*K4 + 144*K1*K4*K5 - 64*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 576*K2**4*K4 - 3024*K2**4 + 160*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1472*K2**2*K3**2 - 568*K2**2*K4**2 + 2400*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 3866*K2**2 - 64*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 736*K2*K3*K5 + 112*K2*K4*K6 + 16*K2*K5*K7 - 2304*K3**2 - 706*K4**2 - 140*K5**2 - 6*K6**2 + 5216
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.143']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81556', 'vk6.81630', 'vk6.81644', 'vk6.81827', 'vk6.81838', 'vk6.82046', 'vk6.82218', 'vk6.82228', 'vk6.82328', 'vk6.82338', 'vk6.82533', 'vk6.82544', 'vk6.82990', 'vk6.83134', 'vk6.83141', 'vk6.83546', 'vk6.83557', 'vk6.83927', 'vk6.84070', 'vk6.84092', 'vk6.84530', 'vk6.84889', 'vk6.84892', 'vk6.85908', 'vk6.85915', 'vk6.86395', 'vk6.86410', 'vk6.86440', 'vk6.86447', 'vk6.88825', 'vk6.89768', 'vk6.89884']
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 O1O2O3O4O5U1O6U2U3U5U6U4
R3 orbit {'O1O2O3O4O5U1O6U2U3U5U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U6U1U3U4O6U5
Gauss code of K* O1O2O3O4O5U6U1U2U5U3O6U4
Gauss code of -K* O1O2O3O4O5U2O6U3U1U4U5U6
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 -3 -1 3 2 3],[ 4 0 1 2 4 3 3],[ 3 -1 0 1 4 2 3],[ 1 -2 -1 0 3 1 2],[-3 -4 -4 -3 0 -1 1],[-2 -3 -2 -1 1 0 1],[-3 -3 -3 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 3 2 -1 -3 -4],[-3 0 1 -1 -3 -4 -4],[-3 -1 0 -1 -2 -3 -3],[-2 1 1 0 -1 -2 -3],[ 1 3 2 1 0 -1 -2],[ 3 4 3 2 1 0 -1],[ 4 4 3 3 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-2,1,3,4,-1,1,3,4,4,1,2,3,3,1,2,3,1,2,1]
Phi over symmetry [-4,-3,-1,2,3,3,0,1,3,3,4,1,3,2,3,2,1,2,0,0,-1]
Phi of -K [-4,-3,-1,2,3,3,0,1,3,3,4,1,3,2,3,2,1,2,0,0,-1]
Phi of K* [-3,-3,-2,1,3,4,-1,0,2,3,4,0,1,2,3,2,3,3,1,1,0]
Phi of -K* [-4,-3,-1,2,3,3,1,2,3,3,4,1,2,3,4,1,2,3,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+86t^4+45t^2+1
Outer characteristic polynomial t^7+134t^5+148t^3+11t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -512*K1**4*K2**2 + 1024*K1**4*K2 - 2528*K1**4 + 128*K1**3*K2**3*K3 - 256*K1**3*K2**2*K3 + 544*K1**3*K2*K3 - 288*K1**3*K3 - 384*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2496*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 9648*K1**2*K2**2 + 160*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 864*K1**2*K2*K4 + 10288*K1**2*K2 - 608*K1**2*K3**2 - 48*K1**2*K4**2 - 6428*K1**2 - 128*K1*K2**4*K3 + 2240*K1*K2**3*K3 + 672*K1*K2**2*K3*K4 - 1760*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 640*K1*K2**2*K5 + 64*K1*K2*K3**3 - 320*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8992*K1*K2*K3 - 32*K1*K2*K4*K5 + 1176*K1*K3*K4 + 144*K1*K4*K5 - 64*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 576*K2**4*K4 - 3024*K2**4 + 160*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1472*K2**2*K3**2 - 568*K2**2*K4**2 + 2400*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 3866*K2**2 - 64*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 736*K2*K3*K5 + 112*K2*K4*K6 + 16*K2*K5*K7 - 2304*K3**2 - 706*K4**2 - 140*K5**2 - 6*K6**2 + 5216
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}]]
If K is slice False
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