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Flat knot 6.77

Min(phi) over symmetries of the knot is: [-4,-3,-1,1,3,4,0,2,2,5,4,1,1,3,2,1,3,3,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.77']
Arrow polynomial of the knot is: 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.77', '6.242', '6.331', '6.862']
Outer characteristic polynomial of the knot is: t^7+138t^5+121t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.77']
2-strand cable arrow polynomial of the knot is: -768*K1**4 - 128*K1**3*K3 - 256*K1**2*K2**4 + 768*K1**2*K2**3 - 4352*K1**2*K2**2 - 384*K1**2*K2*K4 + 5888*K1**2*K2 - 64*K1**2*K3**2 - 32*K1**2*K4**2 - 4640*K1**2 + 1856*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1600*K1*K2**2*K3 - 256*K1*K2**2*K5 + 64*K1*K2*K3*K4**2 - 512*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 6400*K1*K2*K3 + 992*K1*K3*K4 + 112*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**6 - 256*K2**4*K3**2 - 64*K2**4*K4**2 + 256*K2**4*K4 - 1904*K2**4 + 192*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 + 128*K2**2*K3**2*K4 - 1888*K2**2*K3**2 - 64*K2**2*K3*K7 - 432*K2**2*K4**2 + 2352*K2**2*K4 - 48*K2**2*K6**2 - 3456*K2**2 - 64*K2*K3**2*K4 + 960*K2*K3*K5 + 400*K2*K4*K6 + 32*K2*K6*K8 - 64*K3**2*K4**2 + 16*K3**2*K6 - 1984*K3**2 - 832*K4**2 - 96*K5**2 - 104*K6**2 - 8*K8**2 + 3870
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.77']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81597', 'vk6.81681', 'vk6.81686', 'vk6.81745', 'vk6.81758', 'vk6.81987', 'vk6.82284', 'vk6.82402', 'vk6.82434', 'vk6.82516', 'vk6.82715', 'vk6.83205', 'vk6.83612', 'vk6.84193', 'vk6.84381', 'vk6.85980', 'vk6.88179', 'vk6.88753', 'vk6.88780', 'vk6.89119']
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 O1O2O3O4O5O6U2U4U1U6U3U5
R3 orbit {'O1O2O3O4O5O6U2U4U1U6U3U5'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2O3O4O5O6U3U1U5U2U6U4
Gauss code of -K* O1O2O3O4O5O6U3U1U5U2U6U4
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 -3 -4 1 -1 4 3],[ 3 0 -1 3 1 5 3],[ 4 1 0 3 1 4 2],[-1 -3 -3 0 -1 2 1],[ 1 -1 -1 1 0 2 1],[-4 -5 -4 -2 -2 0 0],[-3 -3 -2 -1 -1 0 0]]
Primitive based matrix [[ 0 4 3 1 -1 -3 -4],[-4 0 0 -2 -2 -5 -4],[-3 0 0 -1 -1 -3 -2],[-1 2 1 0 -1 -3 -3],[ 1 2 1 1 0 -1 -1],[ 3 5 3 3 1 0 -1],[ 4 4 2 3 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-3,-1,1,3,4,0,2,2,5,4,1,1,3,2,1,3,3,1,1,1]
Phi over symmetry [-4,-3,-1,1,3,4,0,2,2,5,4,1,1,3,2,1,3,3,1,1,1]
Phi of -K [-4,-3,-1,1,3,4,0,2,2,5,4,1,1,3,2,1,3,3,1,1,1]
Phi of K* [-4,-3,-1,1,3,4,1,1,3,2,4,1,3,3,5,1,1,2,1,2,0]
Phi of -K* [-4,-3,-1,1,3,4,1,1,3,2,4,1,3,3,5,1,1,2,1,2,0]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2+22w^2z+25w
Inner characteristic polynomial t^6+86t^4+15t^2
Outer characteristic polynomial t^7+138t^5+121t^3+4t
Flat arrow polynomial 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 4*K2 + 5
2-strand cable arrow polynomial -768*K1**4 - 128*K1**3*K3 - 256*K1**2*K2**4 + 768*K1**2*K2**3 - 4352*K1**2*K2**2 - 384*K1**2*K2*K4 + 5888*K1**2*K2 - 64*K1**2*K3**2 - 32*K1**2*K4**2 - 4640*K1**2 + 1856*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1600*K1*K2**2*K3 - 256*K1*K2**2*K5 + 64*K1*K2*K3*K4**2 - 512*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 6400*K1*K2*K3 + 992*K1*K3*K4 + 112*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**6 - 256*K2**4*K3**2 - 64*K2**4*K4**2 + 256*K2**4*K4 - 1904*K2**4 + 192*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 + 128*K2**2*K3**2*K4 - 1888*K2**2*K3**2 - 64*K2**2*K3*K7 - 432*K2**2*K4**2 + 2352*K2**2*K4 - 48*K2**2*K6**2 - 3456*K2**2 - 64*K2*K3**2*K4 + 960*K2*K3*K5 + 400*K2*K4*K6 + 32*K2*K6*K8 - 64*K3**2*K4**2 + 16*K3**2*K6 - 1984*K3**2 - 832*K4**2 - 96*K5**2 - 104*K6**2 - 8*K8**2 + 3870
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}]]
If K is slice False
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