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Flat knot 6.864

Min(phi) over symmetries of the knot is: [-3,-1,0,0,1,3,0,2,3,1,2,0,0,0,1,0,0,2,0,3,0]
Flat knots (up to 7 crossings) with same phi are :['6.864']
Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 - 4*K2**2 + 2*K3 + 2*K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.84', '6.123', '6.244', '6.864']
Outer characteristic polynomial of the knot is: t^7+52t^5+83t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.864']
2-strand cable arrow polynomial of the knot is: -416*K1**4 + 192*K1**3*K3*K4 + 160*K1**2*K2 - 576*K1**2*K3**2 - 544*K1**2*K4**2 - 840*K1**2 + 768*K1*K2*K3 + 1856*K1*K3*K4 + 544*K1*K4*K5 - 16*K2**2*K4**2 + 128*K2**2*K4 - 428*K2**2 + 80*K2*K3*K5 + 16*K2*K4*K6 - 32*K3**4 - 96*K3**2*K4**2 + 16*K3**2*K6 - 968*K3**2 + 64*K3*K4*K7 - 48*K4**4 + 32*K4**2*K8 - 888*K4**2 - 200*K5**2 - 4*K6**2 - 8*K7**2 - 4*K8**2 + 1210
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.864']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11197', 'vk6.11208', 'vk6.12392', 'vk6.12409', 'vk6.14518', 'vk6.15738', 'vk6.16161', 'vk6.30786', 'vk6.30804', 'vk6.31992', 'vk6.34082', 'vk6.34199', 'vk6.34480', 'vk6.34518', 'vk6.51926', 'vk6.51944', 'vk6.54164', 'vk6.54368', 'vk6.63603', 'vk6.63614']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4U1U4O5O6U5U3U2U6
R3 orbit {'O1O2O3O4U1U4O5O6U5U3U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3U2U6O5O6U1U4
Gauss code of K* O1O2O3O4U5U3U2U6O5O6U1U4
Gauss code of -K* Same
Diagrammatic symmetry type -
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 0 0 1 -1 3],[ 3 0 3 2 1 0 2],[ 0 -3 0 0 0 0 3],[ 0 -2 0 0 0 0 2],[-1 -1 0 0 0 0 0],[ 1 0 0 0 0 0 1],[-3 -2 -3 -2 0 -1 0]]
Primitive based matrix [[ 0 3 1 0 0 -1 -3],[-3 0 0 -2 -3 -1 -2],[-1 0 0 0 0 0 -1],[ 0 2 0 0 0 0 -2],[ 0 3 0 0 0 0 -3],[ 1 1 0 0 0 0 0],[ 3 2 1 2 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,1,3,0,2,3,1,2,0,0,0,1,0,0,2,0,3,0]
Phi over symmetry [-3,-1,0,0,1,3,0,2,3,1,2,0,0,0,1,0,0,2,0,3,0]
Phi of -K [-3,-1,0,0,1,3,2,0,1,3,4,1,1,2,3,0,1,0,1,1,2]
Phi of K* [-3,-1,0,0,1,3,2,0,1,3,4,1,1,2,3,0,1,0,1,1,2]
Phi of -K* [-3,-1,0,0,1,3,0,2,3,1,2,0,0,0,1,0,0,2,0,3,0]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z+15
Enhanced Jones-Krushkal polynomial -8w^3z+15w^2z+15w
Inner characteristic polynomial t^6+32t^4+27t^2
Outer characteristic polynomial t^7+52t^5+83t^3
Flat arrow polynomial -4*K1*K2 + 2*K1 - 4*K2**2 + 2*K3 + 2*K4 + 3
2-strand cable arrow polynomial -416*K1**4 + 192*K1**3*K3*K4 + 160*K1**2*K2 - 576*K1**2*K3**2 - 544*K1**2*K4**2 - 840*K1**2 + 768*K1*K2*K3 + 1856*K1*K3*K4 + 544*K1*K4*K5 - 16*K2**2*K4**2 + 128*K2**2*K4 - 428*K2**2 + 80*K2*K3*K5 + 16*K2*K4*K6 - 32*K3**4 - 96*K3**2*K4**2 + 16*K3**2*K6 - 968*K3**2 + 64*K3*K4*K7 - 48*K4**4 + 32*K4**2*K8 - 888*K4**2 - 200*K5**2 - 4*K6**2 - 8*K7**2 - 4*K8**2 + 1210
Genus of based matrix 0
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}]]
If K is slice True
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