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Flat knot 6.1755

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,1,2,2,1,0,1,0,1,0,1,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1755']
Arrow polynomial of the knot is: 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.218', '6.554', '6.929', '6.932', '6.1014', '6.1024', '6.1068', '6.1526', '6.1664', '6.1676', '6.1755', '6.1763', '6.2065', '6.2078']
Outer characteristic polynomial of the knot is: t^7+39t^5+95t^3+12t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1755']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1280*K1**4*K2**2 + 1856*K1**4*K2 - 2992*K1**4 + 544*K1**3*K2*K3 + 32*K1**3*K3*K4 - 288*K1**3*K3 - 2048*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 4608*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 11136*K1**2*K2**2 - 768*K1**2*K2*K4 + 9872*K1**2*K2 - 464*K1**2*K3**2 - 32*K1**2*K4**2 - 4948*K1**2 + 2784*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 - 352*K1*K2**2*K5 - 288*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7816*K1*K2*K3 + 1040*K1*K3*K4 + 80*K1*K4*K5 - 64*K2**6 + 352*K2**4*K4 - 3328*K2**4 - 32*K2**3*K6 - 1200*K2**2*K3**2 - 384*K2**2*K4**2 + 2152*K2**2*K4 - 2364*K2**2 + 504*K2*K3*K5 + 96*K2*K4*K6 - 1664*K3**2 - 540*K4**2 - 76*K5**2 - 12*K6**2 + 4122
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1755']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13370', 'vk6.13439', 'vk6.13630', 'vk6.13752', 'vk6.14153', 'vk6.14382', 'vk6.15611', 'vk6.16077', 'vk6.16470', 'vk6.16485', 'vk6.17638', 'vk6.22877', 'vk6.22908', 'vk6.24191', 'vk6.33125', 'vk6.33164', 'vk6.33228', 'vk6.33285', 'vk6.34854', 'vk6.34885', 'vk6.36442', 'vk6.42440', 'vk6.42455', 'vk6.43544', 'vk6.53546', 'vk6.53579', 'vk6.53612', 'vk6.53680', 'vk6.54724', 'vk6.55684', 'vk6.60238', 'vk6.64574']
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 O1O2O3U4U5O4O6U1U6O5U2U3
R3 orbit {'O1O2O3U4U5O4O6U1U6O5U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U2O4U5U3O5O6U4U6
Gauss code of K* O1O2U3O4O5U1U4U5O6O3U6U2
Gauss code of -K* O1O2U3O4O5U4U6O3O6U1U2U5
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 0 2 -1 0 1],[ 2 0 1 2 2 2 1],[ 0 -1 0 1 -1 1 0],[-2 -2 -1 0 -3 -1 0],[ 1 -2 1 3 0 0 1],[ 0 -2 -1 1 0 0 1],[-1 -1 0 0 -1 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -1 -3 -2],[-1 0 0 0 -1 -1 -1],[ 0 1 0 0 1 -1 -1],[ 0 1 1 -1 0 0 -2],[ 1 3 1 1 0 0 -2],[ 2 2 1 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,1,1,3,2,0,1,1,1,-1,1,1,0,2,2]
Phi over symmetry [-2,-1,0,0,1,2,-1,0,1,2,2,1,0,1,0,1,0,1,1,1,1]
Phi of -K [-2,-1,0,0,1,2,-1,0,1,2,2,1,0,1,0,1,0,1,1,1,1]
Phi of K* [-2,-1,0,0,1,2,1,1,1,0,2,0,1,1,2,-1,1,0,0,1,-1]
Phi of -K* [-2,-1,0,0,1,2,2,1,2,1,2,1,0,1,3,1,0,1,1,1,0]
Symmetry type of based matrix c
u-polynomial 0
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+29t^4+67t^2+4
Outer characteristic polynomial t^7+39t^5+95t^3+12t
Flat arrow polynomial 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial 256*K1**4*K2**3 - 1280*K1**4*K2**2 + 1856*K1**4*K2 - 2992*K1**4 + 544*K1**3*K2*K3 + 32*K1**3*K3*K4 - 288*K1**3*K3 - 2048*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 4608*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 11136*K1**2*K2**2 - 768*K1**2*K2*K4 + 9872*K1**2*K2 - 464*K1**2*K3**2 - 32*K1**2*K4**2 - 4948*K1**2 + 2784*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 - 352*K1*K2**2*K5 - 288*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7816*K1*K2*K3 + 1040*K1*K3*K4 + 80*K1*K4*K5 - 64*K2**6 + 352*K2**4*K4 - 3328*K2**4 - 32*K2**3*K6 - 1200*K2**2*K3**2 - 384*K2**2*K4**2 + 2152*K2**2*K4 - 2364*K2**2 + 504*K2*K3*K5 + 96*K2*K4*K6 - 1664*K3**2 - 540*K4**2 - 76*K5**2 - 12*K6**2 + 4122
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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