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Flat knot 6.100

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,1,4,0,1,2,4,4,0,0,1,1,1,2,3,0,2,3]
Flat knots (up to 7 crossings) with same phi are :['6.100']
Arrow polynomial of the knot is: 8*K1**3 + 8*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.100']
Outer characteristic polynomial of the knot is: t^7+102t^5+97t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.100']
2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 768*K1**4*K2 - 1280*K1**4 + 960*K1**3*K2*K3 - 512*K1**3*K3 - 768*K1**2*K2**4 + 1280*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 4864*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 64*K1**2*K2*K3*K5 - 320*K1**2*K2*K4 + 4752*K1**2*K2 - 832*K1**2*K3**2 - 2960*K1**2 + 1984*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 1216*K1*K2**2*K3 - 512*K1*K2**2*K5 + 384*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 5328*K1*K2*K3 + 896*K1*K3*K4 - 192*K2**6 - 384*K2**4*K3**2 - 64*K2**4*K4**2 + 320*K2**4*K4 - 1472*K2**4 + 384*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 128*K2**2*K3**4 + 128*K2**2*K3**2*K6 - 1408*K2**2*K3**2 - 240*K2**2*K4**2 + 1232*K2**2*K4 - 96*K2**2*K5**2 - 48*K2**2*K6**2 - 1936*K2**2 - 192*K2*K3**2*K4 + 816*K2*K3*K5 + 160*K2*K4*K6 - 64*K3**4 + 144*K3**2*K6 - 1584*K3**2 - 388*K4**2 - 128*K5**2 - 64*K6**2 + 2690
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.100']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16961', 'vk6.17203', 'vk6.20853', 'vk6.22254', 'vk6.23363', 'vk6.23658', 'vk6.28312', 'vk6.35412', 'vk6.35832', 'vk6.39934', 'vk6.42023', 'vk6.43159', 'vk6.46480', 'vk6.55115', 'vk6.55375', 'vk6.57664', 'vk6.58847', 'vk6.59816', 'vk6.68393', 'vk6.69716']
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 O1O2O3O4O5O6U2U6U3U4U1U5
R3 orbit {'O1O2O3O4O5O6U2U6U3U4U1U5'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2O3O4O5O6U5U1U3U4U6U2
Gauss code of -K* O1O2O3O4O5O6U5U1U3U4U6U2
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 -4 -1 1 4 1],[ 1 0 -3 0 2 4 1],[ 4 3 0 2 3 4 1],[ 1 0 -2 0 1 2 0],[-1 -2 -3 -1 0 1 0],[-4 -4 -4 -2 -1 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 4 1 1 -1 -1 -4],[-4 0 0 -1 -2 -4 -4],[-1 0 0 0 0 -1 -1],[-1 1 0 0 -1 -2 -3],[ 1 2 0 1 0 0 -2],[ 1 4 1 2 0 0 -3],[ 4 4 1 3 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,-1,1,1,4,0,1,2,4,4,0,0,1,1,1,2,3,0,2,3]
Phi over symmetry [-4,-1,-1,1,1,4,0,1,2,4,4,0,0,1,1,1,2,3,0,2,3]
Phi of -K [-4,-1,-1,1,1,4,0,1,2,4,4,0,0,1,1,1,2,3,0,2,3]
Phi of K* [-4,-1,-1,1,1,4,2,3,1,3,4,0,0,1,2,1,2,4,0,0,1]
Phi of -K* [-4,-1,-1,1,1,4,2,3,1,3,4,0,0,1,2,1,2,4,0,0,1]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2+21w^2z+27w
Inner characteristic polynomial t^6+66t^4+23t^2
Outer characteristic polynomial t^7+102t^5+97t^3+4t
Flat arrow polynomial 8*K1**3 + 8*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 2*K2 + 3
2-strand cable arrow polynomial -512*K1**4*K2**2 + 768*K1**4*K2 - 1280*K1**4 + 960*K1**3*K2*K3 - 512*K1**3*K3 - 768*K1**2*K2**4 + 1280*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 4864*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 64*K1**2*K2*K3*K5 - 320*K1**2*K2*K4 + 4752*K1**2*K2 - 832*K1**2*K3**2 - 2960*K1**2 + 1984*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 1216*K1*K2**2*K3 - 512*K1*K2**2*K5 + 384*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 5328*K1*K2*K3 + 896*K1*K3*K4 - 192*K2**6 - 384*K2**4*K3**2 - 64*K2**4*K4**2 + 320*K2**4*K4 - 1472*K2**4 + 384*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 128*K2**2*K3**4 + 128*K2**2*K3**2*K6 - 1408*K2**2*K3**2 - 240*K2**2*K4**2 + 1232*K2**2*K4 - 96*K2**2*K5**2 - 48*K2**2*K6**2 - 1936*K2**2 - 192*K2*K3**2*K4 + 816*K2*K3*K5 + 160*K2*K4*K6 - 64*K3**4 + 144*K3**2*K6 - 1584*K3**2 - 388*K4**2 - 128*K5**2 - 64*K6**2 + 2690
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
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