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Flat knot 6.813

Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,-1,2,1,2,4,1,0,1,2,0,1,2,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.813']
Arrow polynomial of the knot is: 8*K1**3 - 6*K1**2 - 4*K1*K2 - 4*K1 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.813', '6.1105', '6.1107', '6.1552', '6.1687']
Outer characteristic polynomial of the knot is: t^7+84t^5+72t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.813']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 1408*K1**4*K2 - 4816*K1**4 + 192*K1**3*K2*K3 - 736*K1**3*K3 - 320*K1**2*K2**4 + 1600*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 7888*K1**2*K2**2 - 512*K1**2*K2*K4 + 11488*K1**2*K2 - 400*K1**2*K3**2 - 32*K1**2*K3*K5 - 5176*K1**2 + 352*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6888*K1*K2*K3 + 600*K1*K3*K4 + 32*K1*K4*K5 - 64*K2**6 + 64*K2**4*K4 - 1480*K2**4 - 336*K2**2*K3**2 - 48*K2**2*K4**2 + 1160*K2**2*K4 - 3776*K2**2 + 136*K2*K3*K5 + 16*K2*K4*K6 - 1448*K3**2 - 282*K4**2 - 8*K5**2 + 4408
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.813']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11241', 'vk6.11321', 'vk6.12502', 'vk6.12615', 'vk6.18216', 'vk6.18552', 'vk6.24678', 'vk6.25098', 'vk6.30915', 'vk6.31040', 'vk6.32099', 'vk6.32220', 'vk6.36804', 'vk6.37260', 'vk6.44044', 'vk6.44385', 'vk6.51991', 'vk6.52088', 'vk6.52868', 'vk6.52917', 'vk6.56021', 'vk6.56296', 'vk6.60564', 'vk6.60903', 'vk6.63643', 'vk6.63690', 'vk6.64071', 'vk6.64118', 'vk6.65680', 'vk6.65969', 'vk6.68725', 'vk6.68934']
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 O1O2O3O4U1O5U4U2U6U5O6U3
R3 orbit {'O1O2O3O4U1O5U4U2U6U5O6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2O5U6U5U3U1O6U4
Gauss code of K* O1O2O3O4U3O5U6U2U5U1O6U4
Gauss code of -K* O1O2O3O4U1O5U4U6U3U5O6U2
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 2 0 3 -1],[ 3 0 2 3 1 2 3],[ 1 -2 0 2 0 2 0],[-2 -3 -2 0 -1 2 -3],[ 0 -1 0 1 0 1 -1],[-3 -2 -2 -2 -1 0 -3],[ 1 -3 0 3 1 3 0]]
Primitive based matrix [[ 0 3 2 0 -1 -1 -3],[-3 0 -2 -1 -2 -3 -2],[-2 2 0 -1 -2 -3 -3],[ 0 1 1 0 0 -1 -1],[ 1 2 2 0 0 0 -2],[ 1 3 3 1 0 0 -3],[ 3 2 3 1 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,1,1,3,2,1,2,3,2,1,2,3,3,0,1,1,0,2,3]
Phi over symmetry [-3,-2,0,1,1,3,-1,2,1,2,4,1,0,1,2,0,1,2,0,-1,0]
Phi of -K [-3,-1,-1,0,2,3,-1,0,2,2,4,0,0,0,1,1,1,2,1,2,-1]
Phi of K* [-3,-2,0,1,1,3,-1,2,1,2,4,1,0,1,2,0,1,2,0,-1,0]
Phi of -K* [-3,-1,-1,0,2,3,2,3,1,3,2,0,0,2,2,1,3,3,1,1,2]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 4z^2+25z+35
Enhanced Jones-Krushkal polynomial 4w^3z^2+25w^2z+35w
Inner characteristic polynomial t^6+60t^4+43t^2+4
Outer characteristic polynomial t^7+84t^5+72t^3+11t
Flat arrow polynomial 8*K1**3 - 6*K1**2 - 4*K1*K2 - 4*K1 + 3*K2 + 4
2-strand cable arrow polynomial -192*K1**4*K2**2 + 1408*K1**4*K2 - 4816*K1**4 + 192*K1**3*K2*K3 - 736*K1**3*K3 - 320*K1**2*K2**4 + 1600*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 7888*K1**2*K2**2 - 512*K1**2*K2*K4 + 11488*K1**2*K2 - 400*K1**2*K3**2 - 32*K1**2*K3*K5 - 5176*K1**2 + 352*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6888*K1*K2*K3 + 600*K1*K3*K4 + 32*K1*K4*K5 - 64*K2**6 + 64*K2**4*K4 - 1480*K2**4 - 336*K2**2*K3**2 - 48*K2**2*K4**2 + 1160*K2**2*K4 - 3776*K2**2 + 136*K2*K3*K5 + 16*K2*K4*K6 - 1448*K3**2 - 282*K4**2 - 8*K5**2 + 4408
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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