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Flat knot 6.984

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,1,2,3,4,1,0,2,1,0,1,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.984']
Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 6*K1*K2 + 3*K2 + 2*K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.311', '6.528', '6.536', '6.817', '6.982', '6.984', '6.1284']
Outer characteristic polynomial of the knot is: t^7+46t^5+60t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.984']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 224*K1**4*K2 - 1344*K1**4 + 32*K1**3*K2*K3 + 128*K1**2*K2**3 - 1312*K1**2*K2**2 + 2648*K1**2*K2 - 512*K1**2*K3**2 - 128*K1**2*K4**2 - 1676*K1**2 + 96*K1*K2**3*K3 + 1704*K1*K2*K3 + 880*K1*K3*K4 + 200*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 248*K2**4 - 128*K2**2*K3**2 - 24*K2**2*K4**2 + 256*K2**2*K4 - 1308*K2**2 + 176*K2*K3*K5 + 48*K2*K4*K6 - 804*K3**2 - 462*K4**2 - 152*K5**2 - 36*K6**2 + 1772
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.984']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11526', 'vk6.11859', 'vk6.12876', 'vk6.13185', 'vk6.20355', 'vk6.21696', 'vk6.27659', 'vk6.29203', 'vk6.31297', 'vk6.31694', 'vk6.32455', 'vk6.32872', 'vk6.39089', 'vk6.41343', 'vk6.45845', 'vk6.47510', 'vk6.52297', 'vk6.52563', 'vk6.53141', 'vk6.53447', 'vk6.57226', 'vk6.58451', 'vk6.61840', 'vk6.62975', 'vk6.63802', 'vk6.63936', 'vk6.64248', 'vk6.64446', 'vk6.66835', 'vk6.67703', 'vk6.69475', 'vk6.70197']
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 O1O2O3O4U1U4O5U3O6U2U5U6
R3 orbit {'O1O2O3O4U1U4O5U3O6U2U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U3O5U2O6U1U4
Gauss code of K* O1O2O3U4U1U5U6O4O6U2O5U3
Gauss code of -K* O1O2O3U1O4U2O5O6U5U4U3U6
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 -3 -1 0 1 1 2],[ 3 0 3 2 1 2 1],[ 1 -3 0 0 0 2 2],[ 0 -2 0 0 0 1 1],[-1 -1 0 0 0 0 0],[-1 -2 -2 -1 0 0 1],[-2 -1 -2 -1 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -1 -1 -2 -1],[-1 0 0 0 0 0 -1],[-1 1 0 0 -1 -2 -2],[ 0 1 0 1 0 0 -2],[ 1 2 0 2 0 0 -3],[ 3 1 1 2 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,1,1,2,1,0,0,0,1,1,2,2,0,2,3]
Phi over symmetry [-3,-1,0,1,1,2,-1,1,2,3,4,1,0,2,1,0,1,1,0,0,1]
Phi of -K [-3,-1,0,1,1,2,-1,1,2,3,4,1,0,2,1,0,1,1,0,0,1]
Phi of K* [-2,-1,-1,0,1,3,0,1,1,1,4,0,0,0,2,1,2,3,1,1,-1]
Phi of -K* [-3,-1,0,1,1,2,3,2,1,2,1,0,0,2,2,0,1,1,0,0,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 11z+23
Enhanced Jones-Krushkal polynomial -2w^3z+13w^2z+23w
Inner characteristic polynomial t^6+30t^4+15t^2
Outer characteristic polynomial t^7+46t^5+60t^3
Flat arrow polynomial 4*K1**3 - 6*K1**2 - 6*K1*K2 + 3*K2 + 2*K3 + 4
2-strand cable arrow polynomial -64*K1**6 + 224*K1**4*K2 - 1344*K1**4 + 32*K1**3*K2*K3 + 128*K1**2*K2**3 - 1312*K1**2*K2**2 + 2648*K1**2*K2 - 512*K1**2*K3**2 - 128*K1**2*K4**2 - 1676*K1**2 + 96*K1*K2**3*K3 + 1704*K1*K2*K3 + 880*K1*K3*K4 + 200*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 248*K2**4 - 128*K2**2*K3**2 - 24*K2**2*K4**2 + 256*K2**2*K4 - 1308*K2**2 + 176*K2*K3*K5 + 48*K2*K4*K6 - 804*K3**2 - 462*K4**2 - 152*K5**2 - 36*K6**2 + 1772
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice False
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