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Flat knot 6.70

Min(phi) over symmetries of the knot is: [-4,-3,-2,2,3,4,0,1,3,5,4,0,1,3,2,2,4,3,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.70']
Arrow polynomial of the knot is: -16*K1**4 + 8*K1**3 + 8*K1**2*K2 + 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.69', '6.70']
Outer characteristic polynomial of the knot is: t^7+155t^5+344t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.70']
2-strand cable arrow polynomial of the knot is: -800*K1**4 + 256*K1**2*K2**5 - 2048*K1**2*K2**4 + 1984*K1**2*K2**3 - 4224*K1**2*K2**2 + 3392*K1**2*K2 - 128*K1**2*K3**2 - 2016*K1**2 + 256*K1*K2**5*K3 + 1984*K1*K2**3*K3 + 2848*K1*K2*K3 + 272*K1*K3*K4 + 16*K1*K4*K5 - 768*K2**8 + 768*K2**6*K4 - 2496*K2**6 - 512*K2**4*K3**2 - 192*K2**4*K4**2 + 1536*K2**4*K4 - 1232*K2**4 + 256*K2**3*K3*K5 - 1056*K2**2*K3**2 - 208*K2**2*K4**2 + 848*K2**2*K4 - 64*K2**2*K5**2 + 240*K2**2 + 384*K2*K3*K5 + 16*K2*K5*K7 - 32*K3**4 + 32*K3**2*K6 - 760*K3**2 - 212*K4**2 - 56*K5**2 - 8*K6**2 + 1722
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.70']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81910', 'vk6.81967', 'vk6.82108', 'vk6.82137', 'vk6.82634', 'vk6.82700', 'vk6.82792', 'vk6.82801', 'vk6.83077', 'vk6.83081', 'vk6.83531', 'vk6.84661', 'vk6.84980', 'vk6.85839', 'vk6.86221', 'vk6.88480', 'vk6.89075', 'vk6.89099', 'vk6.89647', 'vk6.90063']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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 O1O2O3O4O5O6U2U3U1U6U4U5
R3 orbit {'O1O2O3O4O5O6U2U3U1U6U4U5'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2O3O4O5O6U3U1U2U5U6U4
Gauss code of -K* O1O2O3O4O5O6U3U1U2U5U6U4
Diagrammatic symmetry type r
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 -4 -2 2 4 3],[ 3 0 -1 1 4 5 3],[ 4 1 0 1 3 4 2],[ 2 -1 -1 0 2 3 1],[-2 -4 -3 -2 0 1 0],[-4 -5 -4 -3 -1 0 0],[-3 -3 -2 -1 0 0 0]]
Primitive based matrix [[ 0 4 3 2 -2 -3 -4],[-4 0 0 -1 -3 -5 -4],[-3 0 0 0 -1 -3 -2],[-2 1 0 0 -2 -4 -3],[ 2 3 1 2 0 -1 -1],[ 3 5 3 4 1 0 -1],[ 4 4 2 3 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-3,-2,2,3,4,0,1,3,5,4,0,1,3,2,2,4,3,1,1,1]
Phi over symmetry [-4,-3,-2,2,3,4,0,1,3,5,4,0,1,3,2,2,4,3,1,1,1]
Phi of -K [-4,-3,-2,2,3,4,0,1,3,5,4,0,1,3,2,2,4,3,1,1,1]
Phi of K* [-4,-3,-2,2,3,4,1,1,3,2,4,1,4,3,5,2,1,3,0,1,0]
Phi of -K* [-4,-3,-2,2,3,4,1,1,3,2,4,1,4,3,5,2,1,3,0,1,0]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial 8w^4z-12w^3z+9w^2z+11w
Inner characteristic polynomial t^6+97t^4+54t^2
Outer characteristic polynomial t^7+155t^5+344t^3
Flat arrow polynomial -16*K1**4 + 8*K1**3 + 8*K1**2*K2 + 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3
2-strand cable arrow polynomial -800*K1**4 + 256*K1**2*K2**5 - 2048*K1**2*K2**4 + 1984*K1**2*K2**3 - 4224*K1**2*K2**2 + 3392*K1**2*K2 - 128*K1**2*K3**2 - 2016*K1**2 + 256*K1*K2**5*K3 + 1984*K1*K2**3*K3 + 2848*K1*K2*K3 + 272*K1*K3*K4 + 16*K1*K4*K5 - 768*K2**8 + 768*K2**6*K4 - 2496*K2**6 - 512*K2**4*K3**2 - 192*K2**4*K4**2 + 1536*K2**4*K4 - 1232*K2**4 + 256*K2**3*K3*K5 - 1056*K2**2*K3**2 - 208*K2**2*K4**2 + 848*K2**2*K4 - 64*K2**2*K5**2 + 240*K2**2 + 384*K2*K3*K5 + 16*K2*K5*K7 - 32*K3**4 + 32*K3**2*K6 - 760*K3**2 - 212*K4**2 - 56*K5**2 - 8*K6**2 + 1722
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}]]
If K is slice False
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