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Flat knot 6.461

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,1,4,1,2,3,2,0,1,2,1,0,2,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.461']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 6*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.172', '6.274', '6.286', '6.423', '6.461']
Outer characteristic polynomial of the knot is: t^7+90t^5+194t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.461']
2-strand cable arrow polynomial of the knot is: -720*K1**4 - 32*K1**3*K3 - 2816*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3136*K1**2*K2**3 - 4160*K1**2*K2**2 - 224*K1**2*K2*K4 + 3432*K1**2*K2 - 80*K1**2*K3**2 - 16*K1**2*K4**2 - 2360*K1**2 + 768*K1*K2**5*K3 + 384*K1*K2**4*K3*K4 - 768*K1*K2**4*K3 - 256*K1*K2**4*K5 - 256*K1*K2**3*K3*K4 + 3168*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 384*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 2904*K1*K2*K3 + 432*K1*K3*K4 + 40*K1*K4*K5 - 736*K2**6 - 896*K2**4*K3**2 - 288*K2**4*K4**2 + 1056*K2**4*K4 - 1784*K2**4 + 448*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 768*K2**2*K3**2 - 152*K2**2*K4**2 + 720*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 460*K2**2 + 176*K2*K3*K5 + 32*K2*K4*K6 - 868*K3**2 - 268*K4**2 - 28*K5**2 - 4*K6**2 + 1858
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.461']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4635', 'vk6.4904', 'vk6.6057', 'vk6.6568', 'vk6.8092', 'vk6.8474', 'vk6.9468', 'vk6.9843', 'vk6.20280', 'vk6.21611', 'vk6.27556', 'vk6.29120', 'vk6.38961', 'vk6.41208', 'vk6.45740', 'vk6.47435', 'vk6.48673', 'vk6.48860', 'vk6.49407', 'vk6.49646', 'vk6.50685', 'vk6.50868', 'vk6.51158', 'vk6.51377', 'vk6.57125', 'vk6.58317', 'vk6.61727', 'vk6.62869', 'vk6.66750', 'vk6.67634', 'vk6.69408', 'vk6.70132']
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 O1O2O3O4O5U6U4O6U1U2U3U5
R3 orbit {'O1O2O3O4O5U6U4O6U1U2U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U3U4U5O6U2U6
Gauss code of K* O1O2O3O4U1U2U3U5U4O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U1U6U2U3U4
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 -3 -1 1 0 4 -1],[ 3 0 1 2 1 4 2],[ 1 -1 0 1 1 3 0],[-1 -2 -1 0 1 2 -2],[ 0 -1 -1 -1 0 0 0],[-4 -4 -3 -2 0 0 -4],[ 1 -2 0 2 0 4 0]]
Primitive based matrix [[ 0 4 1 0 -1 -1 -3],[-4 0 -2 0 -3 -4 -4],[-1 2 0 1 -1 -2 -2],[ 0 0 -1 0 -1 0 -1],[ 1 3 1 1 0 0 -1],[ 1 4 2 0 0 0 -2],[ 3 4 2 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,0,1,1,3,2,0,3,4,4,-1,1,2,2,1,0,1,0,1,2]
Phi over symmetry [-4,-1,0,1,1,3,1,4,1,2,3,2,0,1,2,1,0,2,0,0,1]
Phi of -K [-3,-1,-1,0,1,4,0,1,2,2,3,0,1,0,1,0,1,2,2,4,1]
Phi of K* [-4,-1,0,1,1,3,1,4,1,2,3,2,0,1,2,1,0,2,0,0,1]
Phi of -K* [-3,-1,-1,0,1,4,1,2,1,2,4,0,1,1,3,0,2,4,-1,0,2]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial -6w^4z^2+6w^3z^2-10w^3z+15w^2z+11w
Inner characteristic polynomial t^6+62t^4+102t^2
Outer characteristic polynomial t^7+90t^5+194t^3+6t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 6*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + 2*K2 + 3
2-strand cable arrow polynomial -720*K1**4 - 32*K1**3*K3 - 2816*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3136*K1**2*K2**3 - 4160*K1**2*K2**2 - 224*K1**2*K2*K4 + 3432*K1**2*K2 - 80*K1**2*K3**2 - 16*K1**2*K4**2 - 2360*K1**2 + 768*K1*K2**5*K3 + 384*K1*K2**4*K3*K4 - 768*K1*K2**4*K3 - 256*K1*K2**4*K5 - 256*K1*K2**3*K3*K4 + 3168*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 384*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 2904*K1*K2*K3 + 432*K1*K3*K4 + 40*K1*K4*K5 - 736*K2**6 - 896*K2**4*K3**2 - 288*K2**4*K4**2 + 1056*K2**4*K4 - 1784*K2**4 + 448*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 768*K2**2*K3**2 - 152*K2**2*K4**2 + 720*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 460*K2**2 + 176*K2*K3*K5 + 32*K2*K4*K6 - 868*K3**2 - 268*K4**2 - 28*K5**2 - 4*K6**2 + 1858
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{6}, {4, 5}, {1, 3}, {2}]]
If K is slice False
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