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Flat knot 6.861

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,0,1,3,3,3,0,0,1,2,-1,-1,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.861']
Arrow polynomial of the knot is: 4*K1**3 - 6*K1*K2 + 2*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.426', '6.861', '6.1388']
Outer characteristic polynomial of the knot is: t^7+41t^5+56t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.861']
2-strand cable arrow polynomial of the knot is: 2400*K1**4*K2 - 4640*K1**4 + 1056*K1**3*K2*K3 - 1728*K1**3*K3 - 128*K1**2*K2**4 + 352*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5248*K1**2*K2**2 - 1152*K1**2*K2*K4 + 8112*K1**2*K2 - 1568*K1**2*K3**2 - 320*K1**2*K3*K5 - 96*K1**2*K4**2 - 4288*K1**2 + 352*K1*K2**3*K3 - 704*K1*K2**2*K3 - 448*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 320*K1*K2*K3*K6 + 7480*K1*K2*K3 - 96*K1*K2*K4*K5 + 2696*K1*K3*K4 + 792*K1*K4*K5 + 200*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 592*K2**4 - 32*K2**3*K6 - 384*K2**2*K3**2 - 120*K2**2*K4**2 + 1384*K2**2*K4 - 4052*K2**2 + 1128*K2*K3*K5 + 336*K2*K4*K6 - 32*K3**4 + 184*K3**2*K6 - 2600*K3**2 - 1220*K4**2 - 552*K5**2 - 220*K6**2 + 4450
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.861']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10526', 'vk6.10535', 'vk6.10613', 'vk6.10630', 'vk6.10802', 'vk6.10817', 'vk6.10903', 'vk6.10910', 'vk6.19016', 'vk6.19037', 'vk6.19091', 'vk6.19095', 'vk6.19136', 'vk6.19140', 'vk6.25538', 'vk6.25557', 'vk6.25633', 'vk6.25654', 'vk6.25759', 'vk6.25763', 'vk6.30207', 'vk6.30216', 'vk6.30292', 'vk6.30309', 'vk6.30421', 'vk6.30436', 'vk6.37726', 'vk6.37739', 'vk6.56505', 'vk6.56518', 'vk6.66169', 'vk6.66174']
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 O1O2O3O4U1U4O5O6U3U6U5U2
R3 orbit {'O1O2O3O4U1U4O5O6U3U6U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U6U2O6O5U1U4
Gauss code of K* O1O2O3O4U5U4U1U6O5O6U3U2
Gauss code of -K* O1O2O3O4U3U2O5O6U5U4U1U6
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 3 2 1 1 1],[-1 -3 0 -2 0 1 1],[ 1 -2 2 0 0 2 1],[-1 -1 0 0 0 0 0],[-1 -1 -1 -2 0 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 1 1 0 -2 -3],[-1 -1 0 0 0 -1 -1],[-1 -1 0 0 0 -2 -1],[-1 0 0 0 0 0 -1],[ 1 2 1 2 0 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,-1,0,2,3,0,0,1,1,0,2,1,0,1,2]
Phi over symmetry [-3,-1,1,1,1,1,0,1,3,3,3,0,0,1,2,-1,-1,0,0,0,0]
Phi of -K [-3,-1,1,1,1,1,0,1,3,3,3,0,0,1,2,-1,-1,0,0,0,0]
Phi of K* [-1,-1,-1,-1,1,3,-1,0,0,0,3,0,1,0,1,0,2,3,1,3,0]
Phi of -K* [-3,-1,1,1,1,1,2,1,1,1,3,0,1,2,2,0,0,0,0,-1,-1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+27t^4+18t^2+1
Outer characteristic polynomial t^7+41t^5+56t^3+5t
Flat arrow polynomial 4*K1**3 - 6*K1*K2 + 2*K3 + 1
2-strand cable arrow polynomial 2400*K1**4*K2 - 4640*K1**4 + 1056*K1**3*K2*K3 - 1728*K1**3*K3 - 128*K1**2*K2**4 + 352*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5248*K1**2*K2**2 - 1152*K1**2*K2*K4 + 8112*K1**2*K2 - 1568*K1**2*K3**2 - 320*K1**2*K3*K5 - 96*K1**2*K4**2 - 4288*K1**2 + 352*K1*K2**3*K3 - 704*K1*K2**2*K3 - 448*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 320*K1*K2*K3*K6 + 7480*K1*K2*K3 - 96*K1*K2*K4*K5 + 2696*K1*K3*K4 + 792*K1*K4*K5 + 200*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 592*K2**4 - 32*K2**3*K6 - 384*K2**2*K3**2 - 120*K2**2*K4**2 + 1384*K2**2*K4 - 4052*K2**2 + 1128*K2*K3*K5 + 336*K2*K4*K6 - 32*K3**4 + 184*K3**2*K6 - 2600*K3**2 - 1220*K4**2 - 552*K5**2 - 220*K6**2 + 4450
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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