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Flat knot 6.8

Min(phi) over symmetries of the knot is: [-5,-3,1,1,2,4,1,2,3,5,4,1,2,4,3,0,1,1,1,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.8']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K3 - 2*K1*K4 + K2 + K3 + K5 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.8']
Outer characteristic polynomial of the knot is: t^7+152t^5+185t^3+12t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.8']
2-strand cable arrow polynomial of the knot is: 96*K1**3*K2*K3 + 512*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 2496*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 192*K1**2*K2*K4 + 2696*K1**2*K2 - 768*K1**2*K3**2 - 3104*K1**2 - 128*K1*K2**3*K3*K4 + 992*K1*K2**3*K3 + 672*K1*K2**2*K3*K4 - 960*K1*K2**2*K3 + 224*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 + 384*K1*K2*K3**3 - 832*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 4792*K1*K2*K3 - 32*K1*K2*K4*K5 - 64*K1*K3**2*K5 + 1328*K1*K3*K4 + 352*K1*K4*K5 + 8*K1*K6*K7 + 8*K1*K7*K8 - 2*K10**2 + 8*K10*K2*K8 - 32*K2**4*K4**2 + 320*K2**4*K4 - 1872*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 2400*K2**2*K3**2 + 32*K2**2*K3*K4*K7 - 96*K2**2*K3*K7 - 656*K2**2*K4**2 + 2184*K2**2*K4 - 320*K2**2*K5**2 - 8*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 2268*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 2112*K2*K3*K5 + 176*K2*K4*K6 + 192*K2*K5*K7 + 8*K2*K6*K8 + 8*K2*K7*K9 - 192*K3**4 + 128*K3**2*K6 - 1968*K3**2 + 72*K3*K4*K7 + 8*K3*K5*K8 - 1006*K4**2 - 544*K5**2 - 34*K6**2 - 72*K7**2 - 12*K8**2 + 3200
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.8']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73978', 'vk6.73990', 'vk6.74495', 'vk6.74511', 'vk6.75951', 'vk6.75975', 'vk6.76709', 'vk6.76726', 'vk6.78945', 'vk6.78963', 'vk6.79489', 'vk6.79509', 'vk6.80471', 'vk6.80489', 'vk6.80949', 'vk6.80961', 'vk6.83016', 'vk6.83092', 'vk6.83646', 'vk6.83785', 'vk6.83940', 'vk6.84108', 'vk6.84258', 'vk6.85177', 'vk6.85541', 'vk6.85865', 'vk6.86258', 'vk6.86572', 'vk6.86743', 'vk6.87446', 'vk6.88292', 'vk6.89738']
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 O1O2O3O4O5O6U1U2U5U4U6U3
R3 orbit {'O1O2O3O4O5O6U1U2U5U4U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U4U1U3U2U5U6
Gauss code of K* O1O2O3O4O5O6U1U2U6U4U3U5
Gauss code of -K* O1O2O3O4O5O6U2U4U3U1U5U6
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 -5 -3 2 1 1 4],[ 5 0 1 5 3 2 4],[ 3 -1 0 4 2 1 3],[-2 -5 -4 0 -1 -1 2],[-1 -3 -2 1 0 0 2],[-1 -2 -1 1 0 0 1],[-4 -4 -3 -2 -2 -1 0]]
Primitive based matrix [[ 0 4 2 1 1 -3 -5],[-4 0 -2 -1 -2 -3 -4],[-2 2 0 -1 -1 -4 -5],[-1 1 1 0 0 -1 -2],[-1 2 1 0 0 -2 -3],[ 3 3 4 1 2 0 -1],[ 5 4 5 2 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-1,-1,3,5,2,1,2,3,4,1,1,4,5,0,1,2,2,3,1]
Phi over symmetry [-5,-3,1,1,2,4,1,2,3,5,4,1,2,4,3,0,1,1,1,2,2]
Phi of -K [-5,-3,1,1,2,4,1,3,4,2,5,2,3,1,4,0,0,1,0,2,0]
Phi of K* [-4,-2,-1,-1,3,5,0,1,2,4,5,0,0,1,2,0,2,3,3,4,1]
Phi of -K* [-5,-3,1,1,2,4,1,2,3,5,4,1,2,4,3,0,1,1,1,2,2]
Symmetry type of based matrix c
u-polynomial t^5-t^4+t^3-t^2-2t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial -2w^4z^2+9w^3z^2-4w^3z+28w^2z+21w
Inner characteristic polynomial t^6+96t^4+32t^2
Outer characteristic polynomial t^7+152t^5+185t^3+12t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K3 - 2*K1*K4 + K2 + K3 + K5 + 2
2-strand cable arrow polynomial 96*K1**3*K2*K3 + 512*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 2496*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 192*K1**2*K2*K4 + 2696*K1**2*K2 - 768*K1**2*K3**2 - 3104*K1**2 - 128*K1*K2**3*K3*K4 + 992*K1*K2**3*K3 + 672*K1*K2**2*K3*K4 - 960*K1*K2**2*K3 + 224*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 + 384*K1*K2*K3**3 - 832*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 4792*K1*K2*K3 - 32*K1*K2*K4*K5 - 64*K1*K3**2*K5 + 1328*K1*K3*K4 + 352*K1*K4*K5 + 8*K1*K6*K7 + 8*K1*K7*K8 - 2*K10**2 + 8*K10*K2*K8 - 32*K2**4*K4**2 + 320*K2**4*K4 - 1872*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 2400*K2**2*K3**2 + 32*K2**2*K3*K4*K7 - 96*K2**2*K3*K7 - 656*K2**2*K4**2 + 2184*K2**2*K4 - 320*K2**2*K5**2 - 8*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 2268*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 2112*K2*K3*K5 + 176*K2*K4*K6 + 192*K2*K5*K7 + 8*K2*K6*K8 + 8*K2*K7*K9 - 192*K3**4 + 128*K3**2*K6 - 1968*K3**2 + 72*K3*K4*K7 + 8*K3*K5*K8 - 1006*K4**2 - 544*K5**2 - 34*K6**2 - 72*K7**2 - 12*K8**2 + 3200
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}]]
If K is slice False
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