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Flat knot 6.121

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,2,1,1,3,1,1,1,1,0,1,1,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.121', '7.10419']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.121', '6.125', '6.866', '6.894', '6.936', '6.937']
Outer characteristic polynomial of the knot is: t^7+62t^5+86t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.121']
2-strand cable arrow polynomial of the knot is: -896*K1**4*K2**2 + 1152*K1**4*K2 - 3200*K1**4 + 512*K1**3*K2**3*K3 + 1024*K1**3*K2*K3 - 320*K1**3*K3 + 384*K1**2*K2**5 - 3904*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 5632*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 - 11440*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 8008*K1**2*K2 - 736*K1**2*K3**2 - 128*K1**2*K4**2 - 1980*K1**2 + 640*K1*K2**5*K3 - 768*K1*K2**4*K3 - 256*K1*K2**4*K5 + 4896*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 2080*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 + 96*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6728*K1*K2*K3 + 720*K1*K3*K4 + 168*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1600*K2**6 - 128*K2**5*K6 - 704*K2**4*K3**2 - 64*K2**4*K4**2 + 1440*K2**4*K4 - 4144*K2**4 + 320*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 + 64*K2**2*K3**2*K4 - 1776*K2**2*K3**2 - 32*K2**2*K3*K7 - 408*K2**2*K4**2 + 2488*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 + 134*K2**2 - 32*K2*K3**2*K4 + 576*K2*K3*K5 + 152*K2*K4*K6 - 972*K3**2 - 338*K4**2 - 48*K5**2 - 14*K6**2 + 2360
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.121']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.482', 'vk6.551', 'vk6.582', 'vk6.950', 'vk6.1047', 'vk6.1084', 'vk6.1636', 'vk6.1745', 'vk6.1812', 'vk6.2131', 'vk6.2230', 'vk6.2264', 'vk6.2550', 'vk6.2869', 'vk6.3038', 'vk6.3168', 'vk6.20419', 'vk6.20713', 'vk6.21785', 'vk6.22155', 'vk6.27772', 'vk6.28262', 'vk6.29294', 'vk6.29685', 'vk6.39197', 'vk6.39719', 'vk6.41965', 'vk6.46282', 'vk6.57281', 'vk6.57644', 'vk6.58525', 'vk6.61945']
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 O1O2O3O4O5O6U3U5U6U4U1U2
R3 orbit {'O1O2O3O4O5O6U3U5U6U4U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U6U3U1U2U4
Gauss code of K* O1O2O3O4O5O6U5U6U1U4U2U3
Gauss code of -K* O1O2O3O4O5O6U4U5U3U6U1U2
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 -1 1 -3 1 0 2],[ 1 0 1 -3 1 0 2],[-1 -1 0 -3 1 0 2],[ 3 3 3 0 3 1 2],[-1 -1 -1 -3 0 -1 1],[ 0 0 0 -1 1 0 1],[-2 -2 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 -1 -2 -1 -2 -2],[-1 1 0 -1 -1 -1 -3],[-1 2 1 0 0 -1 -3],[ 0 1 1 0 0 0 -1],[ 1 2 1 1 0 0 -3],[ 3 2 3 3 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,1,2,1,2,2,1,1,1,3,0,1,3,0,1,3]
Phi over symmetry [-3,-1,0,1,1,2,-1,2,1,1,3,1,1,1,1,0,1,1,1,0,-1]
Phi of -K [-3,-1,0,1,1,2,-1,2,1,1,3,1,1,1,1,0,1,1,1,0,-1]
Phi of K* [-2,-1,-1,0,1,3,-1,0,1,1,3,1,1,1,1,0,1,1,1,2,-1]
Phi of -K* [-3,-1,0,1,1,2,3,1,3,3,2,0,1,1,2,0,1,1,1,2,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z^2+16z+21
Enhanced Jones-Krushkal polynomial -2w^4z^2+5w^3z^2-6w^3z+22w^2z+21w
Inner characteristic polynomial t^6+46t^4+31t^2
Outer characteristic polynomial t^7+62t^5+86t^3+7t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -896*K1**4*K2**2 + 1152*K1**4*K2 - 3200*K1**4 + 512*K1**3*K2**3*K3 + 1024*K1**3*K2*K3 - 320*K1**3*K3 + 384*K1**2*K2**5 - 3904*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 5632*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 - 11440*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 8008*K1**2*K2 - 736*K1**2*K3**2 - 128*K1**2*K4**2 - 1980*K1**2 + 640*K1*K2**5*K3 - 768*K1*K2**4*K3 - 256*K1*K2**4*K5 + 4896*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 2080*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 + 96*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6728*K1*K2*K3 + 720*K1*K3*K4 + 168*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1600*K2**6 - 128*K2**5*K6 - 704*K2**4*K3**2 - 64*K2**4*K4**2 + 1440*K2**4*K4 - 4144*K2**4 + 320*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 + 64*K2**2*K3**2*K4 - 1776*K2**2*K3**2 - 32*K2**2*K3*K7 - 408*K2**2*K4**2 + 2488*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 + 134*K2**2 - 32*K2*K3**2*K4 + 576*K2*K3*K5 + 152*K2*K4*K6 - 972*K3**2 - 338*K4**2 - 48*K5**2 - 14*K6**2 + 2360
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice False
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