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Flat knot 6.200

Min(phi) over symmetries of the knot is: [-4,-2,-2,2,3,3,0,1,3,3,5,0,1,1,2,2,2,4,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.200']
Arrow polynomial of the knot is: 12*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1*K3 - 5*K1 + 3*K2 + K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.200']
Outer characteristic polynomial of the knot is: t^7+123t^5+147t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.200']
2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 32*K1**4*K2 - 704*K1**4 + 288*K1**3*K2*K3 - 320*K1**3*K3 + 512*K1**2*K2**5 - 1792*K1**2*K2**4 + 3584*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 7808*K1**2*K2**2 - 448*K1**2*K2*K4 + 7936*K1**2*K2 - 432*K1**2*K3**2 - 32*K1**2*K3*K5 - 5768*K1**2 + 1920*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 + 64*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7328*K1*K2*K3 - 32*K1*K3**2*K5 + 944*K1*K3*K4 + 152*K1*K4*K5 + 8*K1*K5*K6 - 608*K2**6 + 352*K2**4*K4 - 2688*K2**4 + 96*K2**3*K3*K5 - 32*K2**3*K6 - 1376*K2**2*K3**2 - 32*K2**2*K3*K7 - 288*K2**2*K4**2 + 2224*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 2682*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 920*K2*K3*K5 + 80*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 48*K3**2*K6 - 2032*K3**2 + 16*K3*K4*K7 - 518*K4**2 - 184*K5**2 - 14*K6**2 - 8*K7**2 - 2*K8**2 + 4174
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.200']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73710', 'vk6.73829', 'vk6.74189', 'vk6.74797', 'vk6.75639', 'vk6.75825', 'vk6.76346', 'vk6.76866', 'vk6.78618', 'vk6.78813', 'vk6.79222', 'vk6.79693', 'vk6.80252', 'vk6.80390', 'vk6.80695', 'vk6.81065', 'vk6.81607', 'vk6.81797', 'vk6.81923', 'vk6.82174', 'vk6.82296', 'vk6.82644', 'vk6.83198', 'vk6.84059', 'vk6.84207', 'vk6.84687', 'vk6.85004', 'vk6.86022', 'vk6.87759', 'vk6.88201', 'vk6.89403', 'vk6.89602']
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 O1O2O3O4O5U2O6U3U1U6U4U5
R3 orbit {'O1O2O3O4O5U2O6U3U1U6U4U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U2U6U5U3O6U4
Gauss code of K* O1O2O3O4O5U2U6U1U4U5O6U3
Gauss code of -K* O1O2O3O4O5U3O6U1U2U5U6U4
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 -3 -2 2 4 2],[ 3 0 -1 1 4 5 2],[ 3 1 0 1 2 3 1],[ 2 -1 -1 0 2 3 1],[-2 -4 -2 -2 0 1 0],[-4 -5 -3 -3 -1 0 0],[-2 -2 -1 -1 0 0 0]]
Primitive based matrix [[ 0 4 2 2 -2 -3 -3],[-4 0 0 -1 -3 -3 -5],[-2 0 0 0 -1 -1 -2],[-2 1 0 0 -2 -2 -4],[ 2 3 1 2 0 -1 -1],[ 3 3 1 2 1 0 1],[ 3 5 2 4 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-2,2,3,3,0,1,3,3,5,0,1,1,2,2,2,4,1,1,-1]
Phi over symmetry [-4,-2,-2,2,3,3,0,1,3,3,5,0,1,1,2,2,2,4,1,1,-1]
Phi of -K [-3,-3,-2,2,2,4,-1,0,3,4,4,0,1,3,2,2,3,3,0,1,2]
Phi of K* [-4,-2,-2,2,3,3,1,2,3,2,4,0,2,1,3,3,3,4,0,0,-1]
Phi of -K* [-3,-3,-2,2,2,4,-1,1,2,4,5,1,1,2,3,1,2,3,0,0,1]
Symmetry type of based matrix c
u-polynomial -t^4+2t^3-t^2
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial -2w^4z^2+6w^3z^2-2w^3z+23w^2z+27w
Inner characteristic polynomial t^6+77t^4+20t^2
Outer characteristic polynomial t^7+123t^5+147t^3+8t
Flat arrow polynomial 12*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1*K3 - 5*K1 + 3*K2 + K3 + K4 + 3
2-strand cable arrow polynomial -128*K1**4*K2**2 + 32*K1**4*K2 - 704*K1**4 + 288*K1**3*K2*K3 - 320*K1**3*K3 + 512*K1**2*K2**5 - 1792*K1**2*K2**4 + 3584*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 7808*K1**2*K2**2 - 448*K1**2*K2*K4 + 7936*K1**2*K2 - 432*K1**2*K3**2 - 32*K1**2*K3*K5 - 5768*K1**2 + 1920*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 + 64*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7328*K1*K2*K3 - 32*K1*K3**2*K5 + 944*K1*K3*K4 + 152*K1*K4*K5 + 8*K1*K5*K6 - 608*K2**6 + 352*K2**4*K4 - 2688*K2**4 + 96*K2**3*K3*K5 - 32*K2**3*K6 - 1376*K2**2*K3**2 - 32*K2**2*K3*K7 - 288*K2**2*K4**2 + 2224*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 2682*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 920*K2*K3*K5 + 80*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 48*K3**2*K6 - 2032*K3**2 + 16*K3*K4*K7 - 518*K4**2 - 184*K5**2 - 14*K6**2 - 8*K7**2 - 2*K8**2 + 4174
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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