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Flat knot 6.1684

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,0,2,2,2,1,1,2,2,0,1,1,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1684']
Arrow polynomial of the knot is: 4*K1**3 - 14*K1**2 - 8*K1*K2 + K1 + 7*K2 + 3*K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.474', '6.1684', '6.1716', '6.1749', '6.1781']
Outer characteristic polynomial of the knot is: t^7+39t^5+80t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1684']
2-strand cable arrow polynomial of the knot is: 736*K1**4*K2 - 3968*K1**4 + 1120*K1**3*K2*K3 - 1696*K1**3*K3 + 512*K1**2*K2**3 + 288*K1**2*K2**2*K4 - 7312*K1**2*K2**2 + 224*K1**2*K2*K3**2 - 992*K1**2*K2*K4 + 14424*K1**2*K2 - 1888*K1**2*K3**2 - 128*K1**2*K3*K5 - 128*K1**2*K4**2 - 9988*K1**2 + 544*K1*K2**3*K3 - 2048*K1*K2**2*K3 - 352*K1*K2**2*K5 - 672*K1*K2*K3*K4 + 12264*K1*K2*K3 + 2712*K1*K3*K4 + 392*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 1256*K2**4 - 96*K2**3*K6 - 608*K2**2*K3**2 - 128*K2**2*K4**2 + 2144*K2**2*K4 - 7514*K2**2 + 592*K2*K3*K5 + 72*K2*K4*K6 - 3876*K3**2 - 1094*K4**2 - 152*K5**2 - 6*K6**2 + 7684
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1684']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73295', 'vk6.73309', 'vk6.73438', 'vk6.73452', 'vk6.74078', 'vk6.74083', 'vk6.74649', 'vk6.74652', 'vk6.75442', 'vk6.75452', 'vk6.76116', 'vk6.76121', 'vk6.78168', 'vk6.78190', 'vk6.78400', 'vk6.78422', 'vk6.79080', 'vk6.79093', 'vk6.79993', 'vk6.80011', 'vk6.80146', 'vk6.80164', 'vk6.80588', 'vk6.80601', 'vk6.83808', 'vk6.83821', 'vk6.85114', 'vk6.85135', 'vk6.86612', 'vk6.86617', 'vk6.87377', 'vk6.87391']
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 O1O2O3U4O5U1U6U2O4O6U5U3
R3 orbit {'O1O2O3U4O5U1U6U2O4O6U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4O5O6U2U5U3O4U6
Gauss code of K* O1O2O3U4U2O5O6U1U3U6O4U5
Gauss code of -K* O1O2O3U4O5U6U1U3O6O4U2U5
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 1 2 -2 1 0],[ 2 0 1 2 1 1 2],[-1 -1 0 0 -1 -1 0],[-2 -2 0 0 -2 -1 -1],[ 2 -1 1 2 0 2 1],[-1 -1 1 1 -2 0 -1],[ 0 -2 0 1 -1 1 0]]
Primitive based matrix [[ 0 2 1 1 0 -2 -2],[-2 0 0 -1 -1 -2 -2],[-1 0 0 -1 0 -1 -1],[-1 1 1 0 -1 -1 -2],[ 0 1 0 1 0 -2 -1],[ 2 2 1 1 2 0 1],[ 2 2 1 2 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,2,2,0,1,1,2,2,1,0,1,1,1,1,2,2,1,-1]
Phi over symmetry [-2,-2,0,1,1,2,-1,0,2,2,2,1,1,2,2,0,1,1,-1,0,1]
Phi of -K [-2,-2,0,1,1,2,-1,0,2,2,2,1,1,2,2,0,1,1,-1,0,1]
Phi of K* [-2,-1,-1,0,2,2,0,1,1,2,2,1,0,1,2,1,2,2,1,0,-1]
Phi of -K* [-2,-2,0,1,1,2,-1,1,1,2,2,2,1,1,2,0,1,1,-1,0,1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial z^2+22z+41
Enhanced Jones-Krushkal polynomial w^3z^2+22w^2z+41w
Inner characteristic polynomial t^6+25t^4+47t^2+1
Outer characteristic polynomial t^7+39t^5+80t^3+5t
Flat arrow polynomial 4*K1**3 - 14*K1**2 - 8*K1*K2 + K1 + 7*K2 + 3*K3 + 8
2-strand cable arrow polynomial 736*K1**4*K2 - 3968*K1**4 + 1120*K1**3*K2*K3 - 1696*K1**3*K3 + 512*K1**2*K2**3 + 288*K1**2*K2**2*K4 - 7312*K1**2*K2**2 + 224*K1**2*K2*K3**2 - 992*K1**2*K2*K4 + 14424*K1**2*K2 - 1888*K1**2*K3**2 - 128*K1**2*K3*K5 - 128*K1**2*K4**2 - 9988*K1**2 + 544*K1*K2**3*K3 - 2048*K1*K2**2*K3 - 352*K1*K2**2*K5 - 672*K1*K2*K3*K4 + 12264*K1*K2*K3 + 2712*K1*K3*K4 + 392*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 1256*K2**4 - 96*K2**3*K6 - 608*K2**2*K3**2 - 128*K2**2*K4**2 + 2144*K2**2*K4 - 7514*K2**2 + 592*K2*K3*K5 + 72*K2*K4*K6 - 3876*K3**2 - 1094*K4**2 - 152*K5**2 - 6*K6**2 + 7684
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]]
If K is slice False
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