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Flat knot 6.882

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,-1,1,3,2,3,1,2,1,1,1,1,1,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.882']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 6*K1*K2 + K2 + 2*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.362', '6.624', '6.789', '6.859', '6.882', '6.975', '6.989', '6.1048', '6.1057', '6.1158']
Outer characteristic polynomial of the knot is: t^7+61t^5+45t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.882']
2-strand cable arrow polynomial of the knot is: -1536*K1**2*K2**4 + 1088*K1**2*K2**3 - 2544*K1**2*K2**2 + 1256*K1**2*K2 - 16*K1**2*K3**2 - 128*K1**2*K4**2 - 1180*K1**2 + 1568*K1*K2**3*K3 + 2416*K1*K2*K3 + 200*K1*K3*K4 + 216*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 696*K2**4 - 608*K2**2*K3**2 - 56*K2**2*K4**2 + 144*K2**2*K4 - 352*K2**2 + 176*K2*K3*K5 + 32*K2*K4*K6 - 16*K3**4 + 24*K3**2*K6 - 804*K3**2 - 186*K4**2 - 112*K5**2 - 16*K6**2 + 1096
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.882']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70501', 'vk6.70504', 'vk6.70564', 'vk6.70567', 'vk6.70711', 'vk6.70716', 'vk6.70812', 'vk6.70815', 'vk6.70978', 'vk6.70987', 'vk6.71060', 'vk6.71071', 'vk6.71201', 'vk6.71210', 'vk6.71276', 'vk6.71281', 'vk6.71767', 'vk6.72188', 'vk6.74072', 'vk6.74142', 'vk6.74641', 'vk6.74709', 'vk6.76195', 'vk6.76215', 'vk6.77558', 'vk6.79076', 'vk6.79155', 'vk6.80643', 'vk6.81264', 'vk6.87013', 'vk6.87944', 'vk6.89136']
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 O1O2O3O4U2U1O5O6U3U5U6U4
R3 orbit {'O1O2O3O4U2U1O5O6U3U5U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U6U2O5O6U4U3
Gauss code of K* O1O2O3O4U5U6U1U4O6O5U2U3
Gauss code of -K* O1O2O3O4U2U3O5O6U1U4U6U5
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -2 -1 3 0 2],[ 2 0 0 2 3 1 1],[ 2 0 0 1 2 1 1],[ 1 -2 -1 0 3 1 2],[-3 -3 -2 -3 0 -1 1],[ 0 -1 -1 -1 1 0 1],[-2 -1 -1 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 2 0 -1 -2 -2],[-3 0 1 -1 -3 -2 -3],[-2 -1 0 -1 -2 -1 -1],[ 0 1 1 0 -1 -1 -1],[ 1 3 2 1 0 -1 -2],[ 2 2 1 1 1 0 0],[ 2 3 1 1 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,1,2,2,-1,1,3,2,3,1,2,1,1,1,1,1,1,2,0]
Phi over symmetry [-3,-2,0,1,2,2,-1,1,3,2,3,1,2,1,1,1,1,1,1,2,0]
Phi of -K [-2,-2,-1,0,2,3,0,-1,1,3,2,0,1,3,3,0,1,1,1,2,2]
Phi of K* [-3,-2,0,1,2,2,2,2,1,2,3,1,1,3,3,0,1,1,-1,0,0]
Phi of -K* [-2,-2,-1,0,2,3,0,1,1,1,2,2,1,1,3,1,2,3,1,1,-1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 3z+7
Enhanced Jones-Krushkal polynomial -12w^3z+15w^2z+7w
Inner characteristic polynomial t^6+39t^4+12t^2
Outer characteristic polynomial t^7+61t^5+45t^3
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 6*K1*K2 + K2 + 2*K3 + 2
2-strand cable arrow polynomial -1536*K1**2*K2**4 + 1088*K1**2*K2**3 - 2544*K1**2*K2**2 + 1256*K1**2*K2 - 16*K1**2*K3**2 - 128*K1**2*K4**2 - 1180*K1**2 + 1568*K1*K2**3*K3 + 2416*K1*K2*K3 + 200*K1*K3*K4 + 216*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 696*K2**4 - 608*K2**2*K3**2 - 56*K2**2*K4**2 + 144*K2**2*K4 - 352*K2**2 + 176*K2*K3*K5 + 32*K2*K4*K6 - 16*K3**4 + 24*K3**2*K6 - 804*K3**2 - 186*K4**2 - 112*K5**2 - 16*K6**2 + 1096
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice False
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