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Flat knot 6.173

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,1,3,1,2,4,1,0,1,1,0,1,2,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.173']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.173']
Outer characteristic polynomial of the knot is: t^7+96t^5+87t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.173']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 704*K1**4*K2 - 2656*K1**4 + 160*K1**3*K2*K3 - 224*K1**3*K3 + 128*K1**2*K2**5 - 2240*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 4064*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 10304*K1**2*K2**2 - 384*K1**2*K2*K4 + 9360*K1**2*K2 - 96*K1**2*K3**2 - 4168*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**4*K3 - 128*K1*K2**4*K5 + 2656*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1760*K1*K2**2*K3 - 288*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 6992*K1*K2*K3 + 320*K1*K3*K4 + 40*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 928*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 832*K2**4*K4 - 3616*K2**4 + 64*K2**3*K3*K5 - 864*K2**2*K3**2 - 232*K2**2*K4**2 + 2352*K2**2*K4 - 1600*K2**2 + 224*K2*K3*K5 + 32*K2*K4*K6 - 1284*K3**2 - 324*K4**2 - 28*K5**2 + 3514
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.173']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17091', 'vk6.17332', 'vk6.20249', 'vk6.21555', 'vk6.23475', 'vk6.23812', 'vk6.27476', 'vk6.29072', 'vk6.35620', 'vk6.36064', 'vk6.38888', 'vk6.41088', 'vk6.42989', 'vk6.43299', 'vk6.45649', 'vk6.47383', 'vk6.55230', 'vk6.55480', 'vk6.57076', 'vk6.58228', 'vk6.59628', 'vk6.59971', 'vk6.61612', 'vk6.62787', 'vk6.65029', 'vk6.65229', 'vk6.66701', 'vk6.67559', 'vk6.68297', 'vk6.68445', 'vk6.69355', 'vk6.70098']
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 O1O2O3O4O5U1O6U5U3U4U6U2
R3 orbit {'O1O2O3O4O5U1O6U5U3U4U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U6U2U3U1O6U5
Gauss code of K* O1O2O3O4O5U6U5U2U3U1O6U4
Gauss code of -K* O1O2O3O4O5U2O6U5U3U4U1U6
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 -4 1 -1 1 0 3],[ 4 0 4 2 3 1 3],[-1 -4 0 -2 0 -1 3],[ 1 -2 2 0 1 0 3],[-1 -3 0 -1 0 0 2],[ 0 -1 1 0 0 0 1],[-3 -3 -3 -3 -2 -1 0]]
Primitive based matrix [[ 0 3 1 1 0 -1 -4],[-3 0 -2 -3 -1 -3 -3],[-1 2 0 0 0 -1 -3],[-1 3 0 0 -1 -2 -4],[ 0 1 0 1 0 0 -1],[ 1 3 1 2 0 0 -2],[ 4 3 3 4 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,0,1,4,2,3,1,3,3,0,0,1,3,1,2,4,0,1,2]
Phi over symmetry [-4,-1,0,1,1,3,1,3,1,2,4,1,0,1,1,0,1,2,0,-1,0]
Phi of -K [-4,-1,0,1,1,3,1,3,1,2,4,1,0,1,1,0,1,2,0,-1,0]
Phi of K* [-3,-1,-1,0,1,4,-1,0,2,1,4,0,0,0,1,1,1,2,1,3,1]
Phi of -K* [-4,-1,0,1,1,3,2,1,3,4,3,0,1,2,3,0,1,1,0,2,3]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial -2w^4z^2+6w^3z^2-2w^3z+23w^2z+27w
Inner characteristic polynomial t^6+68t^4+33t^2+1
Outer characteristic polynomial t^7+96t^5+87t^3+9t
Flat arrow polynomial -8*K1**4 + 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
2-strand cable arrow polynomial -192*K1**4*K2**2 + 704*K1**4*K2 - 2656*K1**4 + 160*K1**3*K2*K3 - 224*K1**3*K3 + 128*K1**2*K2**5 - 2240*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 4064*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 10304*K1**2*K2**2 - 384*K1**2*K2*K4 + 9360*K1**2*K2 - 96*K1**2*K3**2 - 4168*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**4*K3 - 128*K1*K2**4*K5 + 2656*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1760*K1*K2**2*K3 - 288*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 6992*K1*K2*K3 + 320*K1*K3*K4 + 40*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 928*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 832*K2**4*K4 - 3616*K2**4 + 64*K2**3*K3*K5 - 864*K2**2*K3**2 - 232*K2**2*K4**2 + 2352*K2**2*K4 - 1600*K2**2 + 224*K2*K3*K5 + 32*K2*K4*K6 - 1284*K3**2 - 324*K4**2 - 28*K5**2 + 3514
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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