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Flat knot 6.997

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,1,1,2,3,-1,0,0,0,0,0,0,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.997']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.238', '6.431', '6.945', '6.977', '6.981', '6.997', '6.1050', '6.1070', '6.1098', '6.1376']
Outer characteristic polynomial of the knot is: t^7+42t^5+58t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.997']
2-strand cable arrow polynomial of the knot is: -64*K1**6 - 128*K1**4*K2**2 + 1696*K1**4*K2 - 3136*K1**4 + 64*K1**3*K2*K3 - 672*K1**3*K3 + 1504*K1**2*K2**3 - 7632*K1**2*K2**2 - 736*K1**2*K2*K4 + 9624*K1**2*K2 - 256*K1**2*K3**2 - 64*K1**2*K3*K5 - 32*K1**2*K4**2 - 5908*K1**2 + 416*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 32*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 7640*K1*K2*K3 + 1216*K1*K3*K4 + 224*K1*K4*K5 - 64*K2**6 + 96*K2**4*K4 - 1856*K2**4 - 592*K2**2*K3**2 - 72*K2**2*K4**2 + 1880*K2**2*K4 - 4162*K2**2 + 504*K2*K3*K5 + 40*K2*K4*K6 + 8*K3**2*K6 - 2200*K3**2 - 828*K4**2 - 196*K5**2 - 14*K6**2 + 4978
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.997']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11539', 'vk6.11870', 'vk6.12889', 'vk6.13196', 'vk6.20350', 'vk6.21692', 'vk6.27653', 'vk6.29198', 'vk6.31314', 'vk6.31709', 'vk6.32472', 'vk6.32887', 'vk6.39081', 'vk6.41335', 'vk6.45837', 'vk6.47503', 'vk6.52314', 'vk6.52574', 'vk6.53158', 'vk6.53458', 'vk6.57209', 'vk6.58427', 'vk6.61822', 'vk6.62951', 'vk6.63815', 'vk6.63947', 'vk6.64261', 'vk6.64457', 'vk6.66816', 'vk6.67685', 'vk6.69455', 'vk6.70178']
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 O1O2O3O4U1U5O6U4O5U3U6U2
R3 orbit {'O1O2O3O4U1U5O6U4O5U3U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U2O6U1O5U6U4
Gauss code of K* O1O2O3U4U3U1U5O4O6U2O5U6
Gauss code of -K* O1O2O3U4O5U2O4O6U5U3U1U6
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 0 1 0 1],[ 3 0 3 2 1 2 2],[-1 -3 0 -1 0 -1 1],[ 0 -2 1 0 1 -1 1],[-1 -1 0 -1 0 -1 0],[ 0 -2 1 1 1 0 1],[-1 -2 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 0 -3],[-1 0 1 0 -1 -1 -3],[-1 -1 0 0 -1 -1 -2],[-1 0 0 0 -1 -1 -1],[ 0 1 1 1 0 1 -2],[ 0 1 1 1 -1 0 -2],[ 3 3 2 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,0,3,-1,0,1,1,3,0,1,1,2,1,1,1,-1,2,2]
Phi over symmetry [-3,0,0,1,1,1,1,1,1,2,3,-1,0,0,0,0,0,0,-1,0,0]
Phi of -K [-3,0,0,1,1,1,1,1,1,2,3,-1,0,0,0,0,0,0,-1,0,0]
Phi of K* [-1,-1,-1,0,0,3,-1,0,0,0,2,0,0,0,1,0,0,3,-1,1,1]
Phi of -K* [-3,0,0,1,1,1,2,2,1,2,3,-1,1,1,1,1,1,1,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2-4w^3z+23w^2z+31w
Inner characteristic polynomial t^6+30t^4+30t^2+1
Outer characteristic polynomial t^7+42t^5+58t^3+13t
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -64*K1**6 - 128*K1**4*K2**2 + 1696*K1**4*K2 - 3136*K1**4 + 64*K1**3*K2*K3 - 672*K1**3*K3 + 1504*K1**2*K2**3 - 7632*K1**2*K2**2 - 736*K1**2*K2*K4 + 9624*K1**2*K2 - 256*K1**2*K3**2 - 64*K1**2*K3*K5 - 32*K1**2*K4**2 - 5908*K1**2 + 416*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 32*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 7640*K1*K2*K3 + 1216*K1*K3*K4 + 224*K1*K4*K5 - 64*K2**6 + 96*K2**4*K4 - 1856*K2**4 - 592*K2**2*K3**2 - 72*K2**2*K4**2 + 1880*K2**2*K4 - 4162*K2**2 + 504*K2*K3*K5 + 40*K2*K4*K6 + 8*K3**2*K6 - 2200*K3**2 - 828*K4**2 - 196*K5**2 - 14*K6**2 + 4978
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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