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Flat knot 6.673

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,3,3,2,1,1,2,1,1,1,2,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.673']
Arrow polynomial of the knot is: -10*K1**2 - 2*K1*K2 + K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.574', '6.593', '6.604', '6.649', '6.650', '6.673', '6.690', '6.783', '6.973', '6.985', '6.1033', '6.1035']
Outer characteristic polynomial of the knot is: t^7+60t^5+54t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.673']
2-strand cable arrow polynomial of the knot is: -192*K1**6 - 192*K1**4*K2**2 + 1408*K1**4*K2 - 3568*K1**4 + 288*K1**3*K2*K3 - 768*K1**3*K3 + 608*K1**2*K2**3 - 4192*K1**2*K2**2 - 256*K1**2*K2*K4 + 6944*K1**2*K2 - 432*K1**2*K3**2 - 32*K1**2*K3*K5 - 2792*K1**2 - 480*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 4112*K1*K2*K3 + 568*K1*K3*K4 + 32*K1*K4*K5 - 648*K2**4 - 112*K2**2*K3**2 - 8*K2**2*K4**2 + 664*K2**2*K4 - 2486*K2**2 + 160*K2*K3*K5 + 8*K2*K4*K6 - 1020*K3**2 - 254*K4**2 - 52*K5**2 - 2*K6**2 + 2724
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.673']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11263', 'vk6.11343', 'vk6.12528', 'vk6.12641', 'vk6.17617', 'vk6.18928', 'vk6.19004', 'vk6.19341', 'vk6.19634', 'vk6.24072', 'vk6.24166', 'vk6.25522', 'vk6.25621', 'vk6.26117', 'vk6.26535', 'vk6.30937', 'vk6.31062', 'vk6.32117', 'vk6.32238', 'vk6.36414', 'vk6.37669', 'vk6.37716', 'vk6.43512', 'vk6.44778', 'vk6.52029', 'vk6.52118', 'vk6.52943', 'vk6.56491', 'vk6.56655', 'vk6.65382', 'vk6.66123', 'vk6.66157']
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 O1O2O3O4U2O5U6U4O6U3U1U5
R3 orbit {'O1O2O3O4U2O5U6U4O6U3U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4U2O6U1U6O5U3
Gauss code of K* O1O2O3U2U4U1U5O4U3O6O5U6
Gauss code of -K* O1O2O3U4O5O4U1O6U5U3U6U2
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 -1 -2 0 1 3 -1],[ 1 0 -2 1 2 3 0],[ 2 2 0 2 1 2 1],[ 0 -1 -2 0 1 2 -1],[-1 -2 -1 -1 0 0 -1],[-3 -3 -2 -2 0 0 -3],[ 1 0 -1 1 1 3 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 0 -2 -3 -3 -2],[-1 0 0 -1 -1 -2 -1],[ 0 2 1 0 -1 -1 -2],[ 1 3 1 1 0 0 -1],[ 1 3 2 1 0 0 -2],[ 2 2 1 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,0,2,3,3,2,1,1,2,1,1,1,2,0,1,2]
Phi over symmetry [-3,-1,0,1,1,2,0,2,3,3,2,1,1,2,1,1,1,2,0,1,2]
Phi of -K [-2,-1,-1,0,1,3,-1,0,0,2,3,0,0,0,1,0,1,1,0,1,2]
Phi of K* [-3,-1,0,1,1,2,2,1,1,1,3,0,0,1,2,0,0,0,0,-1,0]
Phi of -K* [-2,-1,-1,0,1,3,1,2,2,1,2,0,1,1,3,1,2,3,1,2,0]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2+19w^2z+31w
Inner characteristic polynomial t^6+44t^4+29t^2+1
Outer characteristic polynomial t^7+60t^5+54t^3+5t
Flat arrow polynomial -10*K1**2 - 2*K1*K2 + K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -192*K1**6 - 192*K1**4*K2**2 + 1408*K1**4*K2 - 3568*K1**4 + 288*K1**3*K2*K3 - 768*K1**3*K3 + 608*K1**2*K2**3 - 4192*K1**2*K2**2 - 256*K1**2*K2*K4 + 6944*K1**2*K2 - 432*K1**2*K3**2 - 32*K1**2*K3*K5 - 2792*K1**2 - 480*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 4112*K1*K2*K3 + 568*K1*K3*K4 + 32*K1*K4*K5 - 648*K2**4 - 112*K2**2*K3**2 - 8*K2**2*K4**2 + 664*K2**2*K4 - 2486*K2**2 + 160*K2*K3*K5 + 8*K2*K4*K6 - 1020*K3**2 - 254*K4**2 - 52*K5**2 - 2*K6**2 + 2724
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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