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Flat knot 6.640

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,2,4,2,1,0,1,1,0,0,1,-1,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.640']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 2*K1*K2 - 2*K1 + 5*K2 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.618', '6.640', '6.736', '6.787']
Outer characteristic polynomial of the knot is: t^7+48t^5+44t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.640']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 864*K1**4*K2 - 2288*K1**4 + 192*K1**3*K2*K3 + 32*K1**3*K3*K4 - 448*K1**3*K3 - 320*K1**2*K2**4 + 1024*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 4688*K1**2*K2**2 - 288*K1**2*K2*K4 + 8168*K1**2*K2 - 240*K1**2*K3**2 - 32*K1**2*K3*K5 - 16*K1**2*K4**2 - 5144*K1**2 + 256*K1*K2**3*K3 - 768*K1*K2**2*K3 - 64*K1*K2**2*K5 + 4472*K1*K2*K3 + 568*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 632*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 608*K2**2*K4 - 3344*K2**2 + 16*K2*K3*K5 - 1140*K3**2 - 238*K4**2 - 4*K5**2 + 3612
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.640']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11220', 'vk6.11299', 'vk6.12485', 'vk6.12596', 'vk6.18229', 'vk6.18565', 'vk6.24699', 'vk6.25115', 'vk6.30894', 'vk6.31017', 'vk6.32082', 'vk6.32201', 'vk6.36823', 'vk6.37287', 'vk6.44067', 'vk6.44407', 'vk6.51978', 'vk6.52073', 'vk6.52863', 'vk6.52910', 'vk6.56022', 'vk6.56297', 'vk6.60569', 'vk6.60910', 'vk6.63638', 'vk6.63683', 'vk6.64070', 'vk6.64115', 'vk6.65687', 'vk6.65980', 'vk6.68736', 'vk6.68945']
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 O1O2O3O4U1O5U4U2O6U5U6U3
R3 orbit {'O1O2O3O4U1O5U4U2O6U5U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U6O5U3U1O6U4
Gauss code of K* O1O2O3U4U5U3U6O4U1O6O5U2
Gauss code of -K* O1O2O3U2O4O5U3O6U5U1U4U6
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 1 1],[ 3 0 2 3 1 2 0],[ 1 -2 0 2 0 2 1],[-2 -3 -2 0 -1 0 1],[ 0 -1 0 1 0 1 1],[-1 -2 -2 0 -1 0 1],[-1 0 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 1 0 -1 -2 -3],[-1 -1 0 -1 -1 -1 0],[-1 0 1 0 -1 -2 -2],[ 0 1 1 1 0 0 -1],[ 1 2 1 2 0 0 -2],[ 3 3 0 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,-1,0,1,2,3,1,1,1,0,1,2,2,0,1,2]
Phi over symmetry [-3,-1,0,1,1,2,0,2,2,4,2,1,0,1,1,0,0,1,-1,1,2]
Phi of -K [-3,-1,0,1,1,2,0,2,2,4,2,1,0,1,1,0,0,1,-1,1,2]
Phi of K* [-2,-1,-1,0,1,3,1,2,1,1,2,1,0,0,2,0,1,4,1,2,0]
Phi of -K* [-3,-1,0,1,1,2,2,1,0,2,3,0,1,2,2,1,1,1,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z^2+22z+33
Enhanced Jones-Krushkal polynomial 3w^3z^2+22w^2z+33w
Inner characteristic polynomial t^6+32t^4+15t^2+1
Outer characteristic polynomial t^7+48t^5+44t^3+6t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 2*K1*K2 - 2*K1 + 5*K2 + 6
2-strand cable arrow polynomial -192*K1**4*K2**2 + 864*K1**4*K2 - 2288*K1**4 + 192*K1**3*K2*K3 + 32*K1**3*K3*K4 - 448*K1**3*K3 - 320*K1**2*K2**4 + 1024*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 4688*K1**2*K2**2 - 288*K1**2*K2*K4 + 8168*K1**2*K2 - 240*K1**2*K3**2 - 32*K1**2*K3*K5 - 16*K1**2*K4**2 - 5144*K1**2 + 256*K1*K2**3*K3 - 768*K1*K2**2*K3 - 64*K1*K2**2*K5 + 4472*K1*K2*K3 + 568*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 632*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 608*K2**2*K4 - 3344*K2**2 + 16*K2*K3*K5 - 1140*K3**2 - 238*K4**2 - 4*K5**2 + 3612
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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