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Flat knot 6.1091

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,0,1,3,3,3,0,1,1,2,-1,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1091']
Arrow polynomial of the knot is: -8*K1**2 - 6*K1*K2 + 3*K1 + 4*K2 + 3*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.248', '6.533', '6.1091']
Outer characteristic polynomial of the knot is: t^7+37t^5+39t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1091', '6.1395']
2-strand cable arrow polynomial of the knot is: -640*K1**6 - 256*K1**4*K2**2 + 2112*K1**4*K2 - 4672*K1**4 + 640*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1440*K1**3*K3 + 640*K1**2*K2**3 - 4112*K1**2*K2**2 - 1056*K1**2*K2*K4 + 9096*K1**2*K2 - 928*K1**2*K3**2 - 32*K1**2*K3*K5 - 240*K1**2*K4**2 - 5368*K1**2 - 864*K1*K2**2*K3 - 64*K1*K2**2*K5 - 288*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7120*K1*K2*K3 + 2552*K1*K3*K4 + 568*K1*K4*K5 + 32*K1*K5*K6 - 352*K2**4 - 144*K2**2*K3**2 - 88*K2**2*K4**2 + 1368*K2**2*K4 - 4994*K2**2 + 576*K2*K3*K5 + 144*K2*K4*K6 + 40*K3**2*K6 - 2880*K3**2 - 1396*K4**2 - 392*K5**2 - 78*K6**2 + 5394
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1091']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10950', 'vk6.10954', 'vk6.10983', 'vk6.10987', 'vk6.12120', 'vk6.12124', 'vk6.12153', 'vk6.12157', 'vk6.13782', 'vk6.13800', 'vk6.14212', 'vk6.14233', 'vk6.14661', 'vk6.14680', 'vk6.14853', 'vk6.14875', 'vk6.15819', 'vk6.15840', 'vk6.31820', 'vk6.31832', 'vk6.33620', 'vk6.33630', 'vk6.33653', 'vk6.33663', 'vk6.51780', 'vk6.51800', 'vk6.52645', 'vk6.52665', 'vk6.53808', 'vk6.53826', 'vk6.54230', 'vk6.54251']
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 O1O2O3O4U1U4O5U2U5O6U3U6
R3 orbit {'O1O2O3O4U1U4O5U2U5O6U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2O5U6U3O6U1U4
Gauss code of K* O1O2U3O4O3U5O6O5U1U4U6U2
Gauss code of -K* O1O2U1O3O4U3O5O6U5U2U4U6
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 1 1 1 1],[ 3 0 2 3 1 1 1],[ 1 -2 0 2 0 1 1],[-1 -3 -2 0 0 0 1],[-1 -1 0 0 0 0 0],[-1 -1 -1 0 0 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 1 0 0 -2 -3],[-1 -1 0 0 0 -1 -1],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 -1 -1],[ 1 2 1 0 1 0 -2],[ 3 3 1 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,1,3,-1,0,0,2,3,0,0,1,1,0,0,1,1,1,2]
Phi over symmetry [-3,-1,1,1,1,1,0,1,3,3,3,0,1,1,2,-1,0,0,0,0,0]
Phi of -K [-3,-1,1,1,1,1,0,1,3,3,3,0,1,1,2,-1,0,0,0,0,0]
Phi of K* [-1,-1,-1,-1,1,3,-1,0,0,1,3,0,0,0,1,0,1,3,2,3,0]
Phi of -K* [-3,-1,1,1,1,1,2,1,1,1,3,0,1,1,2,0,0,0,0,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial z^2+22z+41
Enhanced Jones-Krushkal polynomial w^3z^2+22w^2z+41w
Inner characteristic polynomial t^6+23t^4+11t^2+1
Outer characteristic polynomial t^7+37t^5+39t^3+7t
Flat arrow polynomial -8*K1**2 - 6*K1*K2 + 3*K1 + 4*K2 + 3*K3 + 5
2-strand cable arrow polynomial -640*K1**6 - 256*K1**4*K2**2 + 2112*K1**4*K2 - 4672*K1**4 + 640*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1440*K1**3*K3 + 640*K1**2*K2**3 - 4112*K1**2*K2**2 - 1056*K1**2*K2*K4 + 9096*K1**2*K2 - 928*K1**2*K3**2 - 32*K1**2*K3*K5 - 240*K1**2*K4**2 - 5368*K1**2 - 864*K1*K2**2*K3 - 64*K1*K2**2*K5 - 288*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7120*K1*K2*K3 + 2552*K1*K3*K4 + 568*K1*K4*K5 + 32*K1*K5*K6 - 352*K2**4 - 144*K2**2*K3**2 - 88*K2**2*K4**2 + 1368*K2**2*K4 - 4994*K2**2 + 576*K2*K3*K5 + 144*K2*K4*K6 + 40*K3**2*K6 - 2880*K3**2 - 1396*K4**2 - 392*K5**2 - 78*K6**2 + 5394
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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