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Flat knot 6.525

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,-1,2,4,2,3,2,2,1,2,1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.525']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 - 2*K2**2 + 5*K2 + K3 + K4 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.525']
Outer characteristic polynomial of the knot is: t^7+57t^5+62t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.525']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 224*K1**4*K2 - 864*K1**4 + 128*K1**3*K2**3*K3 + 448*K1**3*K2*K3 - 64*K1**3*K3 - 832*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3456*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 8848*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 8320*K1**2*K2 - 416*K1**2*K3**2 - 96*K1**2*K4**2 - 6108*K1**2 - 640*K1*K2**4*K3 - 128*K1*K2**3*K3*K4 + 2688*K1*K2**3*K3 + 640*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 288*K1*K2**2*K5 + 128*K1*K2*K3**3 - 704*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7784*K1*K2*K3 - 32*K1*K2*K4*K5 + 1472*K1*K3*K4 + 328*K1*K4*K5 + 40*K1*K5*K6 - 64*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 640*K2**4*K4 - 3200*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 + 128*K2**2*K3**2*K4 - 1664*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 472*K2**2*K4**2 + 2400*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 3078*K2**2 - 32*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 856*K2*K3*K5 - 32*K2*K4**2*K6 + 144*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 - 32*K3**2*K4**2 + 40*K3**2*K6 - 2248*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 926*K4**2 - 228*K5**2 - 66*K6**2 - 2*K8**2 + 4686
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.525']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3619', 'vk6.3696', 'vk6.3889', 'vk6.4004', 'vk6.7045', 'vk6.7092', 'vk6.7269', 'vk6.7378', 'vk6.17693', 'vk6.17740', 'vk6.24244', 'vk6.24303', 'vk6.36535', 'vk6.36610', 'vk6.43645', 'vk6.43750', 'vk6.48247', 'vk6.48320', 'vk6.48405', 'vk6.48428', 'vk6.50007', 'vk6.50046', 'vk6.50131', 'vk6.50152', 'vk6.55725', 'vk6.55780', 'vk6.60301', 'vk6.60382', 'vk6.65433', 'vk6.65460', 'vk6.68565', 'vk6.68592']
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 O1O2O3O4O5U2U4U5O6U1U6U3
R3 orbit {'O1O2O3O4O5U2U4U5O6U1U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U6U5O6U1U2U4
Gauss code of K* O1O2O3U4O5O4O6U5U1U6U2U3
Gauss code of -K* O1O2O3U2O4O5O6U4U5U1U6U3
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 -2 -3 2 0 2 1],[ 2 0 -2 3 0 2 1],[ 3 2 0 3 1 2 0],[-2 -3 -3 0 -1 1 0],[ 0 0 -1 1 0 1 0],[-2 -2 -2 -1 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 2 1 0 -2 -3],[-2 0 1 0 -1 -3 -3],[-2 -1 0 0 -1 -2 -2],[-1 0 0 0 0 -1 0],[ 0 1 1 0 0 0 -1],[ 2 3 2 1 0 0 -2],[ 3 3 2 0 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,2,3,-1,0,1,3,3,0,1,2,2,0,1,0,0,1,2]
Phi over symmetry [-3,-2,0,1,2,2,-1,2,4,2,3,2,2,1,2,1,1,1,1,1,-1]
Phi of -K [-3,-2,0,1,2,2,-1,2,4,2,3,2,2,1,2,1,1,1,1,1,-1]
Phi of K* [-2,-2,-1,0,2,3,-1,1,1,2,3,1,1,1,2,1,2,4,2,2,-1]
Phi of -K* [-3,-2,0,1,2,2,2,1,0,2,3,0,1,2,3,0,1,1,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z^2+20z+29
Enhanced Jones-Krushkal polynomial 3w^3z^2-4w^3z+24w^2z+29w
Inner characteristic polynomial t^6+35t^4+23t^2
Outer characteristic polynomial t^7+57t^5+62t^3+8t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 - 2*K2**2 + 5*K2 + K3 + K4 + 7
2-strand cable arrow polynomial -192*K1**4*K2**2 + 224*K1**4*K2 - 864*K1**4 + 128*K1**3*K2**3*K3 + 448*K1**3*K2*K3 - 64*K1**3*K3 - 832*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3456*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 8848*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 8320*K1**2*K2 - 416*K1**2*K3**2 - 96*K1**2*K4**2 - 6108*K1**2 - 640*K1*K2**4*K3 - 128*K1*K2**3*K3*K4 + 2688*K1*K2**3*K3 + 640*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 288*K1*K2**2*K5 + 128*K1*K2*K3**3 - 704*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7784*K1*K2*K3 - 32*K1*K2*K4*K5 + 1472*K1*K3*K4 + 328*K1*K4*K5 + 40*K1*K5*K6 - 64*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 640*K2**4*K4 - 3200*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 + 128*K2**2*K3**2*K4 - 1664*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 472*K2**2*K4**2 + 2400*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 3078*K2**2 - 32*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 856*K2*K3*K5 - 32*K2*K4**2*K6 + 144*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 - 32*K3**2*K4**2 + 40*K3**2*K6 - 2248*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 926*K4**2 - 228*K5**2 - 66*K6**2 - 2*K8**2 + 4686
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}]]
If K is slice False
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