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Flat knot 6.1696

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,2,3,-2,1,0,0,1,1,0,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1696']
Arrow polynomial of the knot is: 8*K1**3 - 10*K1**2 - 8*K1*K2 - 2*K1 + 5*K2 + 2*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.543', '6.1656', '6.1696', '6.1770', '6.1772', '6.1794']
Outer characteristic polynomial of the knot is: t^7+23t^5+54t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1696']
2-strand cable arrow polynomial of the knot is: -1024*K1**4*K2**2 + 1760*K1**4*K2 - 2528*K1**4 + 672*K1**3*K2*K3 - 416*K1**3*K3 - 1344*K1**2*K2**4 + 3840*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 11008*K1**2*K2**2 - 704*K1**2*K2*K4 + 9872*K1**2*K2 - 320*K1**2*K3**2 - 5488*K1**2 + 1952*K1*K2**3*K3 - 1792*K1*K2**2*K3 - 416*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 8544*K1*K2*K3 + 792*K1*K3*K4 + 104*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 2936*K2**4 - 64*K2**3*K6 - 832*K2**2*K3**2 - 128*K2**2*K4**2 + 2376*K2**2*K4 - 3404*K2**2 + 592*K2*K3*K5 + 48*K2*K4*K6 - 2004*K3**2 - 646*K4**2 - 156*K5**2 - 4*K6**2 + 4628
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1696']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13366', 'vk6.13431', 'vk6.13622', 'vk6.13748', 'vk6.14158', 'vk6.14393', 'vk6.15624', 'vk6.16084', 'vk6.16463', 'vk6.16480', 'vk6.17639', 'vk6.22866', 'vk6.22899', 'vk6.24192', 'vk6.33121', 'vk6.33156', 'vk6.33220', 'vk6.33281', 'vk6.34847', 'vk6.34880', 'vk6.36443', 'vk6.42437', 'vk6.42454', 'vk6.43545', 'vk6.53550', 'vk6.53587', 'vk6.53620', 'vk6.53684', 'vk6.54714', 'vk6.55685', 'vk6.60239', 'vk6.64581']
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 O1O2O3U1U2O4O5U3U5O6U4U6
R3 orbit {'O1O2O3U1U2O4O5U3U5O6U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4U6U1O6O5U2U3
Gauss code of K* O1O2U3O4O3U5U6U1O5O6U4U2
Gauss code of -K* O1O2U1O3O4U3U2O5O6U4U5U6
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 -2 0 0 0 1 1],[ 2 0 1 2 1 1 0],[ 0 -1 0 1 1 1 0],[ 0 -2 -1 0 2 1 1],[ 0 -1 -1 -2 0 0 1],[-1 -1 -1 -1 0 0 0],[-1 0 0 -1 -1 0 0]]
Primitive based matrix [[ 0 1 1 0 0 0 -2],[-1 0 0 0 -1 -1 0],[-1 0 0 -1 0 -1 -1],[ 0 0 1 0 1 1 -1],[ 0 1 0 -1 0 -2 -1],[ 0 1 1 -1 2 0 -2],[ 2 0 1 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,0,2,0,0,1,1,0,1,0,1,1,-1,-1,1,2,1,2]
Phi over symmetry [-2,0,0,0,1,1,0,1,1,2,3,-2,1,0,0,1,1,0,0,1,0]
Phi of -K [-2,0,0,0,1,1,0,1,1,2,3,-2,1,0,0,1,1,0,0,1,0]
Phi of K* [-1,-1,0,0,0,2,0,0,0,1,2,0,1,0,3,-1,2,0,1,1,1]
Phi of -K* [-2,0,0,0,1,1,1,1,2,0,1,-1,-2,1,0,1,0,1,1,1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 4z^2+23z+31
Enhanced Jones-Krushkal polynomial 4w^3z^2-2w^3z+25w^2z+31w
Inner characteristic polynomial t^6+17t^4+11t^2
Outer characteristic polynomial t^7+23t^5+54t^3+9t
Flat arrow polynomial 8*K1**3 - 10*K1**2 - 8*K1*K2 - 2*K1 + 5*K2 + 2*K3 + 6
2-strand cable arrow polynomial -1024*K1**4*K2**2 + 1760*K1**4*K2 - 2528*K1**4 + 672*K1**3*K2*K3 - 416*K1**3*K3 - 1344*K1**2*K2**4 + 3840*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 11008*K1**2*K2**2 - 704*K1**2*K2*K4 + 9872*K1**2*K2 - 320*K1**2*K3**2 - 5488*K1**2 + 1952*K1*K2**3*K3 - 1792*K1*K2**2*K3 - 416*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 8544*K1*K2*K3 + 792*K1*K3*K4 + 104*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 2936*K2**4 - 64*K2**3*K6 - 832*K2**2*K3**2 - 128*K2**2*K4**2 + 2376*K2**2*K4 - 3404*K2**2 + 592*K2*K3*K5 + 48*K2*K4*K6 - 2004*K3**2 - 646*K4**2 - 156*K5**2 - 4*K6**2 + 4628
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
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