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Flat knot 6.1148

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,-1,2,2,3,4,0,2,2,1,0,2,1,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.1148']
Arrow polynomial of the knot is: 12*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 4*K1*K3 - 6*K1 + 4*K2 + K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1148']
Outer characteristic polynomial of the knot is: t^7+43t^5+132t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1148']
2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 64*K1**4*K2 - 608*K1**4 + 256*K1**3*K2*K3 - 224*K1**3*K3 - 1920*K1**2*K2**4 + 3040*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7888*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 6792*K1**2*K2 - 288*K1**2*K3**2 - 4672*K1**2 + 640*K1*K2**5*K3 - 256*K1*K2**4*K3 - 256*K1*K2**4*K5 + 4544*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2080*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 512*K1*K2**2*K5 + 128*K1*K2*K3**3 - 416*K1*K2*K3*K4 + 7000*K1*K2*K3 - 96*K1*K2*K4*K5 + 736*K1*K3*K4 + 120*K1*K4*K5 + 16*K1*K5*K6 - 608*K2**6 - 768*K2**4*K3**2 - 32*K2**4*K4**2 + 512*K2**4*K4 - 3904*K2**4 + 512*K2**3*K3*K5 + 64*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 2608*K2**2*K3**2 - 64*K2**2*K3*K7 - 264*K2**2*K4**2 - 32*K2**2*K4*K8 + 2456*K2**2*K4 - 128*K2**2*K5**2 - 16*K2**2*K6**2 - 1360*K2**2 - 128*K2*K3**2*K4 + 1056*K2*K3*K5 + 136*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 32*K3**2*K6 - 1796*K3**2 - 468*K4**2 - 148*K5**2 - 24*K6**2 - 2*K8**2 + 3524
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1148']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70415', 'vk6.70424', 'vk6.70428', 'vk6.70441', 'vk6.70445', 'vk6.70454', 'vk6.70456', 'vk6.70535', 'vk6.70540', 'vk6.70616', 'vk6.70771', 'vk6.70780', 'vk6.70854', 'vk6.70861', 'vk6.70874', 'vk6.70886', 'vk6.70891', 'vk6.70903', 'vk6.70906', 'vk6.71018', 'vk6.71027', 'vk6.71128', 'vk6.71135', 'vk6.71257', 'vk6.71845', 'vk6.72284', 'vk6.72296', 'vk6.76661', 'vk6.76678', 'vk6.77632', 'vk6.87959', 'vk6.89212']
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 O1O2O3O4U1U3U4O5O6U2U5U6
R3 orbit {'O1O2O3O4U1U3U4O5O6U2U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U3O5O6U1U2U4
Gauss code of K* O1O2O3U4U5O6O4O5U1U6U2U3
Gauss code of -K* O1O2O3U1U2O4O5O6U4U5U3U6
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 -1 0 2 0 2],[ 3 0 3 1 2 1 1],[ 1 -3 0 -1 1 1 2],[ 0 -1 1 0 1 0 0],[-2 -2 -1 -1 0 0 0],[ 0 -1 -1 0 0 0 1],[-2 -1 -2 0 0 -1 0]]
Primitive based matrix [[ 0 2 2 0 0 -1 -3],[-2 0 0 0 -1 -1 -2],[-2 0 0 -1 0 -2 -1],[ 0 0 1 0 0 -1 -1],[ 0 1 0 0 0 1 -1],[ 1 1 2 1 -1 0 -3],[ 3 2 1 1 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,1,3,0,0,1,1,2,1,0,2,1,0,1,1,-1,1,3]
Phi over symmetry [-3,-1,0,0,2,2,-1,2,2,3,4,0,2,2,1,0,2,1,1,2,0]
Phi of -K [-3,-1,0,0,2,2,-1,2,2,3,4,0,2,2,1,0,2,1,1,2,0]
Phi of K* [-2,-2,0,0,1,3,0,1,2,1,4,2,1,2,3,0,0,2,2,2,-1]
Phi of -K* [-3,-1,0,0,2,2,3,1,1,1,2,-1,1,2,1,0,0,1,1,0,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial -2w^4z^2+7w^3z^2-2w^3z+24w^2z+25w
Inner characteristic polynomial t^6+25t^4+42t^2+1
Outer characteristic polynomial t^7+43t^5+132t^3+9t
Flat arrow polynomial 12*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 4*K1*K3 - 6*K1 + 4*K2 + K4 + 4
2-strand cable arrow polynomial -64*K1**4*K2**2 + 64*K1**4*K2 - 608*K1**4 + 256*K1**3*K2*K3 - 224*K1**3*K3 - 1920*K1**2*K2**4 + 3040*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7888*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 6792*K1**2*K2 - 288*K1**2*K3**2 - 4672*K1**2 + 640*K1*K2**5*K3 - 256*K1*K2**4*K3 - 256*K1*K2**4*K5 + 4544*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2080*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 512*K1*K2**2*K5 + 128*K1*K2*K3**3 - 416*K1*K2*K3*K4 + 7000*K1*K2*K3 - 96*K1*K2*K4*K5 + 736*K1*K3*K4 + 120*K1*K4*K5 + 16*K1*K5*K6 - 608*K2**6 - 768*K2**4*K3**2 - 32*K2**4*K4**2 + 512*K2**4*K4 - 3904*K2**4 + 512*K2**3*K3*K5 + 64*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 2608*K2**2*K3**2 - 64*K2**2*K3*K7 - 264*K2**2*K4**2 - 32*K2**2*K4*K8 + 2456*K2**2*K4 - 128*K2**2*K5**2 - 16*K2**2*K6**2 - 1360*K2**2 - 128*K2*K3**2*K4 + 1056*K2*K3*K5 + 136*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 32*K3**2*K6 - 1796*K3**2 - 468*K4**2 - 148*K5**2 - 24*K6**2 - 2*K8**2 + 3524
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}], [{6}, {3, 5}, {4}, {1, 2}]]
If K is slice False
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