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Flat knot 6.135

Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,-1,-1,2,2,2,0,2,2,2,1,1,1,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.135']
Arrow polynomial of the knot is: -8*K1**2 - 4*K1*K2 - 4*K1*K3 + 2*K1 + 6*K2 + 2*K3 + 2*K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.135']
Outer characteristic polynomial of the knot is: t^7+53t^5+53t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.135']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 128*K1**4*K2 - 2336*K1**4 - 448*K1**3*K3 - 2688*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 6224*K1**2*K2 - 1984*K1**2*K3**2 - 160*K1**2*K4**2 - 4584*K1**2 - 768*K1*K2**2*K3 + 128*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 320*K1*K2*K3*K6 + 7200*K1*K2*K3 + 2864*K1*K3*K4 + 272*K1*K4*K5 + 80*K1*K5*K6 - 160*K2**4 - 704*K2**2*K3**2 - 64*K2**2*K3*K7 - 48*K2**2*K4**2 + 848*K2**2*K4 - 48*K2**2*K6**2 - 4108*K2**2 - 64*K2*K3**2*K4 + 880*K2*K3*K5 + 256*K2*K4*K6 + 64*K2*K5*K7 + 32*K2*K6*K8 - 160*K3**4 + 256*K3**2*K6 - 2864*K3**2 + 32*K3*K4*K7 - 1052*K4**2 - 248*K5**2 - 156*K6**2 - 32*K7**2 - 4*K8**2 + 4438
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.135']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4782', 'vk6.5117', 'vk6.6347', 'vk6.6775', 'vk6.8301', 'vk6.8751', 'vk6.9671', 'vk6.9980', 'vk6.21032', 'vk6.22454', 'vk6.28474', 'vk6.40252', 'vk6.42173', 'vk6.46750', 'vk6.48806', 'vk6.49021', 'vk6.49837', 'vk6.51496', 'vk6.58973', 'vk6.69807']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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 O1O2O3O4O5O6U4U6U2U5U1U3
R3 orbit {'O1O2O3O4O5O6U4U6U2U5U1U3'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2O3O4O5O6U5U3U6U1U4U2
Gauss code of -K* O1O2O3O4O5O6U5U3U6U1U4U2
Diagrammatic symmetry type r
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 -2 2 -2 2 1],[ 1 0 -1 2 -2 2 1],[ 2 1 0 2 -1 2 1],[-2 -2 -2 0 -2 1 1],[ 2 2 1 2 0 2 1],[-2 -2 -2 -1 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 2 2 1 -1 -2 -2],[-2 0 1 1 -2 -2 -2],[-2 -1 0 0 -2 -2 -2],[-1 -1 0 0 -1 -1 -1],[ 1 2 2 1 0 -1 -2],[ 2 2 2 1 1 0 -1],[ 2 2 2 1 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,2,2,-1,-1,2,2,2,0,2,2,2,1,1,1,1,2,1]
Phi over symmetry [-2,-2,-1,1,2,2,-1,-1,2,2,2,0,2,2,2,1,1,1,1,2,1]
Phi of -K [-2,-2,-1,1,2,2,-1,-1,2,2,2,0,2,2,2,1,1,1,1,2,1]
Phi of K* [-2,-2,-1,1,2,2,-1,1,1,2,2,2,1,2,2,1,2,2,-1,0,1]
Phi of -K* [-2,-2,-1,1,2,2,-1,1,1,2,2,2,1,2,2,1,2,2,-1,0,1]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 17z+35
Enhanced Jones-Krushkal polynomial 17w^2z+35w
Inner characteristic polynomial t^6+35t^4+15t^2
Outer characteristic polynomial t^7+53t^5+53t^3+4t
Flat arrow polynomial -8*K1**2 - 4*K1*K2 - 4*K1*K3 + 2*K1 + 6*K2 + 2*K3 + 2*K4 + 5
2-strand cable arrow polynomial -128*K1**6 + 128*K1**4*K2 - 2336*K1**4 - 448*K1**3*K3 - 2688*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 6224*K1**2*K2 - 1984*K1**2*K3**2 - 160*K1**2*K4**2 - 4584*K1**2 - 768*K1*K2**2*K3 + 128*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 320*K1*K2*K3*K6 + 7200*K1*K2*K3 + 2864*K1*K3*K4 + 272*K1*K4*K5 + 80*K1*K5*K6 - 160*K2**4 - 704*K2**2*K3**2 - 64*K2**2*K3*K7 - 48*K2**2*K4**2 + 848*K2**2*K4 - 48*K2**2*K6**2 - 4108*K2**2 - 64*K2*K3**2*K4 + 880*K2*K3*K5 + 256*K2*K4*K6 + 64*K2*K5*K7 + 32*K2*K6*K8 - 160*K3**4 + 256*K3**2*K6 - 2864*K3**2 + 32*K3*K4*K7 - 1052*K4**2 - 248*K5**2 - 156*K6**2 - 32*K7**2 - 4*K8**2 + 4438
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}]]
If K is slice False
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