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Flat knot 6.1170

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,0,1,2,3,3,0,1,1,1,0,-1,0,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1170']
Arrow polynomial of the knot is: 4*K1**3 - 16*K1**2 - 10*K1*K2 + 2*K1 + 8*K2 + 4*K3 + 9
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1170']
Outer characteristic polynomial of the knot is: t^7+43t^5+43t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1170']
2-strand cable arrow polynomial of the knot is: -768*K1**6 - 512*K1**4*K2**2 + 2784*K1**4*K2 - 6432*K1**4 + 800*K1**3*K2*K3 - 1248*K1**3*K3 + 384*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6976*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 800*K1**2*K2*K4 + 13384*K1**2*K2 - 1472*K1**2*K3**2 - 160*K1**2*K3*K5 - 128*K1**2*K4**2 - 7160*K1**2 + 96*K1*K2**3*K3 - 1440*K1*K2**2*K3 - 128*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 10360*K1*K2*K3 + 2864*K1*K3*K4 + 504*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 688*K2**4 - 32*K2**3*K6 - 512*K2**2*K3**2 - 136*K2**2*K4**2 + 1848*K2**2*K4 - 6864*K2**2 + 768*K2*K3*K5 + 112*K2*K4*K6 - 64*K3**4 + 64*K3**2*K6 - 3600*K3**2 - 1388*K4**2 - 320*K5**2 - 40*K6**2 + 7122
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1170']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10963', 'vk6.10967', 'vk6.10994', 'vk6.10998', 'vk6.12133', 'vk6.12137', 'vk6.12164', 'vk6.12168', 'vk6.13787', 'vk6.13803', 'vk6.14221', 'vk6.14225', 'vk6.14668', 'vk6.14672', 'vk6.14862', 'vk6.14878', 'vk6.15828', 'vk6.15832', 'vk6.31835', 'vk6.31839', 'vk6.33627', 'vk6.33643', 'vk6.33658', 'vk6.33674', 'vk6.51791', 'vk6.51795', 'vk6.52658', 'vk6.52662', 'vk6.53797', 'vk6.53813', 'vk6.54223', 'vk6.54227']
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 O1O2O3O4U1U5U3O5O6U4U6U2
R3 orbit {'O1O2O3O4U1U5U3O5O6U4U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U1O5O6U2U6U4
Gauss code of K* O1O2O3U2U4O5O4O6U1U6U3U5
Gauss code of -K* O1O2O3U2U4O5O4O6U3U5U1U6
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 -3 1 1 1 -1 1],[ 3 0 3 1 2 2 1],[-1 -3 0 0 0 -2 1],[-1 -1 0 0 0 -1 1],[-1 -2 0 0 0 -1 1],[ 1 -2 2 1 1 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 1 0 0 -1 -1],[-1 -1 0 -1 -1 -1 -1],[-1 0 1 0 0 -1 -2],[-1 0 1 0 0 -2 -3],[ 1 1 1 1 2 0 -2],[ 3 1 1 2 3 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,1,3,-1,0,0,1,1,1,1,1,1,0,1,2,2,3,2]
Phi over symmetry [-3,-1,1,1,1,1,0,1,2,3,3,0,1,1,1,0,-1,0,-1,0,1]
Phi of -K [-3,-1,1,1,1,1,0,1,2,3,3,0,1,1,1,0,-1,0,-1,0,1]
Phi of K* [-1,-1,-1,-1,1,3,-1,-1,-1,1,3,0,0,0,1,0,1,2,1,3,0]
Phi of -K* [-3,-1,1,1,1,1,2,1,1,2,3,1,1,1,2,-1,-1,-1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial z^2+22z+41
Enhanced Jones-Krushkal polynomial w^3z^2+22w^2z+41w
Inner characteristic polynomial t^6+29t^4+17t^2+1
Outer characteristic polynomial t^7+43t^5+43t^3+5t
Flat arrow polynomial 4*K1**3 - 16*K1**2 - 10*K1*K2 + 2*K1 + 8*K2 + 4*K3 + 9
2-strand cable arrow polynomial -768*K1**6 - 512*K1**4*K2**2 + 2784*K1**4*K2 - 6432*K1**4 + 800*K1**3*K2*K3 - 1248*K1**3*K3 + 384*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6976*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 800*K1**2*K2*K4 + 13384*K1**2*K2 - 1472*K1**2*K3**2 - 160*K1**2*K3*K5 - 128*K1**2*K4**2 - 7160*K1**2 + 96*K1*K2**3*K3 - 1440*K1*K2**2*K3 - 128*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 10360*K1*K2*K3 + 2864*K1*K3*K4 + 504*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 688*K2**4 - 32*K2**3*K6 - 512*K2**2*K3**2 - 136*K2**2*K4**2 + 1848*K2**2*K4 - 6864*K2**2 + 768*K2*K3*K5 + 112*K2*K4*K6 - 64*K3**4 + 64*K3**2*K6 - 3600*K3**2 - 1388*K4**2 - 320*K5**2 - 40*K6**2 + 7122
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}]]
If K is slice False
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