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Flat knot 6.252

Min(phi) over symmetries of the knot is: [-4,-2,-1,2,2,3,1,1,3,4,3,0,2,2,2,2,1,1,-1,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.252']
Arrow polynomial of the knot is: 12*K1**3 - 8*K1**2 - 10*K1*K2 - 2*K1*K3 - 4*K1 + 5*K2 + 2*K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.252']
Outer characteristic polynomial of the knot is: t^7+107t^5+107t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.252']
2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 128*K1**4*K2 - 608*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 256*K1**3*K2*K3 - 288*K1**3*K3 - 1408*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3744*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 9152*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 8432*K1**2*K2 - 336*K1**2*K3**2 - 32*K1**2*K3*K5 - 6328*K1**2 - 256*K1*K2**4*K3 + 3328*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 2272*K1*K2**2*K3 - 480*K1*K2**2*K5 + 64*K1*K2*K3**3 - 448*K1*K2*K3*K4 + 8216*K1*K2*K3 + 1168*K1*K3*K4 + 192*K1*K4*K5 + 8*K1*K5*K6 - 96*K2**6 + 480*K2**4*K4 - 3856*K2**4 + 64*K2**3*K3*K5 - 64*K2**3*K6 - 1920*K2**2*K3**2 - 392*K2**2*K4**2 + 2696*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 2832*K2**2 - 64*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 960*K2*K3*K5 + 152*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 48*K3**2*K6 - 2336*K3**2 + 16*K3*K4*K7 - 822*K4**2 - 232*K5**2 - 40*K6**2 - 16*K7**2 - 2*K8**2 + 4774
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.252']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73268', 'vk6.73409', 'vk6.74020', 'vk6.74566', 'vk6.75172', 'vk6.75406', 'vk6.76044', 'vk6.76770', 'vk6.78133', 'vk6.78372', 'vk6.78997', 'vk6.79558', 'vk6.79970', 'vk6.80125', 'vk6.80525', 'vk6.80989', 'vk6.81870', 'vk6.82161', 'vk6.82186', 'vk6.82586', 'vk6.83581', 'vk6.83771', 'vk6.84044', 'vk6.84598', 'vk6.84926', 'vk6.85581', 'vk6.85716', 'vk6.85947', 'vk6.86736', 'vk6.87660', 'vk6.88931', 'vk6.89969']
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 O1O2O3O4O5U1U2O6U3U5U6U4
R3 orbit {'O1O2O3O4O5U1U2O6U3U5U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U6U1U3O6U4U5
Gauss code of K* O1O2O3O4U5U6U1U4U2O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U3U1U4U5U6
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 -2 -1 3 2 2],[ 4 0 1 2 4 3 2],[ 2 -1 0 1 3 2 2],[ 1 -2 -1 0 3 1 2],[-3 -4 -3 -3 0 -1 1],[-2 -3 -2 -1 1 0 1],[-2 -2 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 2 2 -1 -2 -4],[-3 0 1 -1 -3 -3 -4],[-2 -1 0 -1 -2 -2 -2],[-2 1 1 0 -1 -2 -3],[ 1 3 2 1 0 -1 -2],[ 2 3 2 2 1 0 -1],[ 4 4 2 3 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-2,1,2,4,-1,1,3,3,4,1,2,2,2,1,2,3,1,2,1]
Phi over symmetry [-4,-2,-1,2,2,3,1,1,3,4,3,0,2,2,2,2,1,1,-1,0,2]
Phi of -K [-4,-2,-1,2,2,3,1,1,3,4,3,0,2,2,2,2,1,1,-1,0,2]
Phi of K* [-3,-2,-2,1,2,4,0,2,1,2,3,1,2,2,3,1,2,4,0,1,1]
Phi of -K* [-4,-2,-1,2,2,3,1,2,2,3,4,1,2,2,3,2,1,3,-1,-1,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2-4w^3z+25w^2z+27w
Inner characteristic polynomial t^6+69t^4+24t^2+1
Outer characteristic polynomial t^7+107t^5+107t^3+9t
Flat arrow polynomial 12*K1**3 - 8*K1**2 - 10*K1*K2 - 2*K1*K3 - 4*K1 + 5*K2 + 2*K3 + K4 + 5
2-strand cable arrow polynomial -128*K1**4*K2**2 + 128*K1**4*K2 - 608*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 256*K1**3*K2*K3 - 288*K1**3*K3 - 1408*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3744*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 9152*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 8432*K1**2*K2 - 336*K1**2*K3**2 - 32*K1**2*K3*K5 - 6328*K1**2 - 256*K1*K2**4*K3 + 3328*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 2272*K1*K2**2*K3 - 480*K1*K2**2*K5 + 64*K1*K2*K3**3 - 448*K1*K2*K3*K4 + 8216*K1*K2*K3 + 1168*K1*K3*K4 + 192*K1*K4*K5 + 8*K1*K5*K6 - 96*K2**6 + 480*K2**4*K4 - 3856*K2**4 + 64*K2**3*K3*K5 - 64*K2**3*K6 - 1920*K2**2*K3**2 - 392*K2**2*K4**2 + 2696*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 2832*K2**2 - 64*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 960*K2*K3*K5 + 152*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 48*K3**2*K6 - 2336*K3**2 + 16*K3*K4*K7 - 822*K4**2 - 232*K5**2 - 40*K6**2 - 16*K7**2 - 2*K8**2 + 4774
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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