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Flat knot 6.509

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,0,1,2,5,3,0,0,1,0,1,2,2,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.509']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 4*K1**2*K2 + 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.509']
Outer characteristic polynomial of the knot is: t^7+104t^5+42t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.509']
2-strand cable arrow polynomial of the knot is: -176*K1**4 - 768*K1**2*K2**6 + 1408*K1**2*K2**5 - 2240*K1**2*K2**4 + 992*K1**2*K2**3 - 1232*K1**2*K2**2 + 1432*K1**2*K2 - 48*K1**2*K3**2 - 16*K1**2*K4**2 - 1188*K1**2 + 640*K1*K2**5*K3 + 672*K1*K2**3*K3 + 792*K1*K2*K3 + 192*K1*K3*K4 + 40*K1*K4*K5 + 16*K1*K5*K6 - 128*K2**8 + 128*K2**6*K4 - 800*K2**6 - 192*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 + 360*K2**4 + 32*K2**3*K3*K5 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 104*K2**2*K4 - 416*K2**2 + 32*K2*K3*K5 - 288*K3**2 - 122*K4**2 - 36*K5**2 - 8*K6**2 + 824
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.509']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4637', 'vk6.4908', 'vk6.6065', 'vk6.6572', 'vk6.8096', 'vk6.8482', 'vk6.9472', 'vk6.9845', 'vk6.20281', 'vk6.21610', 'vk6.27557', 'vk6.29119', 'vk6.38962', 'vk6.41207', 'vk6.45741', 'vk6.47434', 'vk6.48671', 'vk6.48856', 'vk6.49399', 'vk6.49642', 'vk6.50681', 'vk6.50860', 'vk6.51154', 'vk6.51375', 'vk6.57126', 'vk6.58316', 'vk6.61728', 'vk6.62868', 'vk6.66751', 'vk6.67633', 'vk6.69409', 'vk6.70131']
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 O1O2O3O4O5U1U6U5O6U2U3U4
R3 orbit {'O1O2O3O4O5U1U6U5O6U2U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U3U4O6U1U6U5
Gauss code of K* O1O2O3U2O4O5O6U1U4U5U6U3
Gauss code of -K* O1O2O3U4O5O4O6U5U1U2U3U6
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -4 -1 1 3 2 -1],[ 4 0 2 3 4 1 3],[ 1 -2 0 1 2 1 0],[-1 -3 -1 0 1 1 -2],[-3 -4 -2 -1 0 1 -4],[-2 -1 -1 -1 -1 0 -2],[ 1 -3 0 2 4 2 0]]
Primitive based matrix [[ 0 3 2 1 -1 -1 -4],[-3 0 1 -1 -2 -4 -4],[-2 -1 0 -1 -1 -2 -1],[-1 1 1 0 -1 -2 -3],[ 1 2 1 1 0 0 -2],[ 1 4 2 2 0 0 -3],[ 4 4 1 3 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,1,4,-1,1,2,4,4,1,1,2,1,1,2,3,0,2,3]
Phi over symmetry [-4,-1,-1,1,2,3,0,1,2,5,3,0,0,1,0,1,2,2,0,1,2]
Phi of -K [-4,-1,-1,1,2,3,0,1,2,5,3,0,0,1,0,1,2,2,0,1,2]
Phi of K* [-3,-2,-1,1,1,4,2,1,0,2,3,0,1,2,5,0,1,2,0,0,1]
Phi of -K* [-4,-1,-1,1,2,3,2,3,3,1,4,0,1,1,2,2,2,4,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 3z+7
Enhanced Jones-Krushkal polynomial 8w^4z-12w^3z+7w^2z+7w
Inner characteristic polynomial t^6+72t^4+9t^2
Outer characteristic polynomial t^7+104t^5+42t^3
Flat arrow polynomial -8*K1**4 + 4*K1**3 + 4*K1**2*K2 + 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2
2-strand cable arrow polynomial -176*K1**4 - 768*K1**2*K2**6 + 1408*K1**2*K2**5 - 2240*K1**2*K2**4 + 992*K1**2*K2**3 - 1232*K1**2*K2**2 + 1432*K1**2*K2 - 48*K1**2*K3**2 - 16*K1**2*K4**2 - 1188*K1**2 + 640*K1*K2**5*K3 + 672*K1*K2**3*K3 + 792*K1*K2*K3 + 192*K1*K3*K4 + 40*K1*K4*K5 + 16*K1*K5*K6 - 128*K2**8 + 128*K2**6*K4 - 800*K2**6 - 192*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 + 360*K2**4 + 32*K2**3*K3*K5 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 104*K2**2*K4 - 416*K2**2 + 32*K2*K3*K5 - 288*K3**2 - 122*K4**2 - 36*K5**2 - 8*K6**2 + 824
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}]]
If K is slice False
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