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Flat knot 6.53

Min(phi) over symmetries of the knot is: [-5,-1,1,1,2,2,2,1,5,3,4,0,2,1,2,0,0,0,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.53']
Arrow polynomial of the knot is: 4*K1**3 - 6*K1*K2 - 2*K1*K4 + 3*K3 + K5 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.53']
Outer characteristic polynomial of the knot is: t^7+102t^5+102t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.53']
2-strand cable arrow polynomial of the knot is: 768*K1**4*K2 - 2880*K1**4 + 1344*K1**3*K2*K3 - 1088*K1**3*K3 - 128*K1**2*K2**4 + 256*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4064*K1**2*K2**2 - 1152*K1**2*K2*K4 + 6688*K1**2*K2 - 2880*K1**2*K3**2 - 192*K1**2*K3*K5 - 160*K1**2*K4**2 - 4648*K1**2 + 640*K1*K2**3*K3 - 832*K1*K2**2*K3 - 576*K1*K2**2*K5 + 512*K1*K2*K3**3 - 576*K1*K2*K3*K4 - 256*K1*K2*K3*K6 + 8160*K1*K2*K3 - 64*K1*K2*K4*K5 - 64*K1*K2*K4*K7 + 3392*K1*K3*K4 + 816*K1*K4*K5 + 64*K1*K5*K6 + 32*K1*K6*K7 - 2*K10**2 + 8*K10*K2*K8 - 32*K2**6 + 96*K2**4*K4 - 752*K2**4 + 64*K2**3*K3*K5 - 32*K2**3*K6 + 128*K2**2*K3**2*K4 - 1184*K2**2*K3**2 - 64*K2**2*K3*K7 - 88*K2**2*K4**2 - 32*K2**2*K4*K8 + 1632*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K8**2 - 4368*K2**2 - 192*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1680*K2*K3*K5 + 272*K2*K4*K6 + 128*K2*K5*K7 + 40*K2*K6*K8 - 320*K3**4 - 64*K3**2*K4**2 + 336*K3**2*K6 - 2952*K3**2 + 80*K3*K4*K7 + 32*K3*K5*K8 + 16*K4**2*K8 - 1388*K4**2 - 720*K5**2 - 174*K6**2 - 64*K7**2 - 36*K8**2 + 4846
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.53']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20155', 'vk6.20163', 'vk6.21442', 'vk6.21449', 'vk6.27279', 'vk6.27287', 'vk6.28939', 'vk6.28947', 'vk6.38696', 'vk6.38712', 'vk6.40890', 'vk6.47272', 'vk6.47281', 'vk6.56980', 'vk6.56991', 'vk6.58131', 'vk6.62683', 'vk6.67463', 'vk6.70039', 'vk6.70043']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3O4O5O6U1U6U3U5U4U2
R3 orbit {'O1O2O3O4O5O6U1U6U3U5U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U3U2U4U1U6
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U5U3U2U4U1U6
Diagrammatic symmetry type +
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 1 -1 2 2 1],[ 5 0 5 2 4 3 1],[-1 -5 0 -2 1 1 0],[ 1 -2 2 0 2 1 0],[-2 -4 -1 -2 0 0 0],[-2 -3 -1 -1 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 2 1 1 -1 -5],[-2 0 0 0 -1 -1 -3],[-2 0 0 0 -1 -2 -4],[-1 0 0 0 0 0 -1],[-1 1 1 0 0 -2 -5],[ 1 1 2 0 2 0 -2],[ 5 3 4 1 5 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,-1,1,5,0,0,1,1,3,0,1,2,4,0,0,1,2,5,2]
Phi over symmetry [-5,-1,1,1,2,2,2,1,5,3,4,0,2,1,2,0,0,0,1,1,0]
Phi of -K [-5,-1,1,1,2,2,2,1,5,3,4,0,2,1,2,0,0,0,1,1,0]
Phi of K* [-2,-2,-1,-1,1,5,0,0,1,1,3,0,1,2,4,0,0,1,2,5,2]
Phi of -K* [-5,-1,1,1,2,2,2,1,5,3,4,0,2,1,2,0,0,0,1,1,0]
Symmetry type of based matrix +
u-polynomial t^5-2t^2-t
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+66t^4+26t^2+1
Outer characteristic polynomial t^7+102t^5+102t^3+5t
Flat arrow polynomial 4*K1**3 - 6*K1*K2 - 2*K1*K4 + 3*K3 + K5 + 1
2-strand cable arrow polynomial 768*K1**4*K2 - 2880*K1**4 + 1344*K1**3*K2*K3 - 1088*K1**3*K3 - 128*K1**2*K2**4 + 256*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4064*K1**2*K2**2 - 1152*K1**2*K2*K4 + 6688*K1**2*K2 - 2880*K1**2*K3**2 - 192*K1**2*K3*K5 - 160*K1**2*K4**2 - 4648*K1**2 + 640*K1*K2**3*K3 - 832*K1*K2**2*K3 - 576*K1*K2**2*K5 + 512*K1*K2*K3**3 - 576*K1*K2*K3*K4 - 256*K1*K2*K3*K6 + 8160*K1*K2*K3 - 64*K1*K2*K4*K5 - 64*K1*K2*K4*K7 + 3392*K1*K3*K4 + 816*K1*K4*K5 + 64*K1*K5*K6 + 32*K1*K6*K7 - 2*K10**2 + 8*K10*K2*K8 - 32*K2**6 + 96*K2**4*K4 - 752*K2**4 + 64*K2**3*K3*K5 - 32*K2**3*K6 + 128*K2**2*K3**2*K4 - 1184*K2**2*K3**2 - 64*K2**2*K3*K7 - 88*K2**2*K4**2 - 32*K2**2*K4*K8 + 1632*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K8**2 - 4368*K2**2 - 192*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1680*K2*K3*K5 + 272*K2*K4*K6 + 128*K2*K5*K7 + 40*K2*K6*K8 - 320*K3**4 - 64*K3**2*K4**2 + 336*K3**2*K6 - 2952*K3**2 + 80*K3*K4*K7 + 32*K3*K5*K8 + 16*K4**2*K8 - 1388*K4**2 - 720*K5**2 - 174*K6**2 - 64*K7**2 - 36*K8**2 + 4846
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}]]
If K is slice False
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