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Flat knot 6.975

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,0,2,3,1,0,1,2,1,-1,0,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.975']
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+42t^5+74t^3+14t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.975']
2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 608*K1**4*K2 - 688*K1**4 + 128*K1**3*K2**3*K3 - 256*K1**3*K2**2*K3 + 1056*K1**3*K2*K3 - 544*K1**3*K3 - 384*K1**2*K2**4 + 1600*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 7376*K1**2*K2**2 - 512*K1**2*K2*K4 + 5840*K1**2*K2 - 352*K1**2*K3**2 - 3936*K1**2 + 1984*K1*K2**3*K3 - 928*K1*K2**2*K3 - 448*K1*K2**2*K5 + 96*K1*K2*K3**3 + 6736*K1*K2*K3 + 376*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 1672*K2**4 - 32*K2**3*K6 - 816*K2**2*K3**2 - 24*K2**2*K4**2 + 1112*K2**2*K4 - 2188*K2**2 + 224*K2*K3*K5 + 24*K2*K4*K6 - 16*K3**4 - 1684*K3**2 - 214*K4**2 - 4*K5**2 - 4*K6**2 + 2964
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.975']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4184', 'vk6.4263', 'vk6.5426', 'vk6.5544', 'vk6.7543', 'vk6.7623', 'vk6.9053', 'vk6.9132', 'vk6.18247', 'vk6.18584', 'vk6.24723', 'vk6.25138', 'vk6.36853', 'vk6.37318', 'vk6.44082', 'vk6.44423', 'vk6.48504', 'vk6.48583', 'vk6.49192', 'vk6.49300', 'vk6.50291', 'vk6.50363', 'vk6.51058', 'vk6.51089', 'vk6.56042', 'vk6.56318', 'vk6.60595', 'vk6.60940', 'vk6.65712', 'vk6.66008', 'vk6.68753', 'vk6.68963']
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 O1O2O3O4U1U2O5U4O6U5U6U3
R3 orbit {'O1O2O3O4U1U2O5U4O6U5U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U6O5U1O6U3U4
Gauss code of K* O1O2O3U4U5U3U6O4O5U1O6U2
Gauss code of -K* O1O2O3U2O4U3O5O6U4U1U5U6
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 2 1 0 1],[ 3 0 1 3 2 1 0],[ 1 -1 0 2 1 1 0],[-2 -3 -2 0 -1 0 1],[-1 -2 -1 1 0 1 1],[ 0 -1 -1 0 -1 0 1],[-1 0 0 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 1 -1 0 -2 -3],[-1 -1 0 -1 -1 0 0],[-1 1 1 0 1 -1 -2],[ 0 0 1 -1 0 -1 -1],[ 1 2 0 1 1 0 -1],[ 3 3 0 2 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,-1,1,0,2,3,1,1,0,0,-1,1,2,1,1,1]
Phi over symmetry [-3,-1,0,1,1,2,1,1,0,2,3,1,0,1,2,1,-1,0,-1,-1,1]
Phi of -K [-3,-1,0,1,1,2,1,2,2,4,2,0,1,2,1,2,0,2,-1,0,2]
Phi of K* [-2,-1,-1,0,1,3,0,2,2,1,2,1,2,1,2,0,2,4,0,2,1]
Phi of -K* [-3,-1,0,1,1,2,1,1,0,2,3,1,0,1,2,1,-1,0,-1,-1,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+26t^4+17t^2+1
Outer characteristic polynomial t^7+42t^5+74t^3+14t
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 6*K1*K2 + K2 + 2*K3 + 2
2-strand cable arrow polynomial -512*K1**4*K2**2 + 608*K1**4*K2 - 688*K1**4 + 128*K1**3*K2**3*K3 - 256*K1**3*K2**2*K3 + 1056*K1**3*K2*K3 - 544*K1**3*K3 - 384*K1**2*K2**4 + 1600*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 7376*K1**2*K2**2 - 512*K1**2*K2*K4 + 5840*K1**2*K2 - 352*K1**2*K3**2 - 3936*K1**2 + 1984*K1*K2**3*K3 - 928*K1*K2**2*K3 - 448*K1*K2**2*K5 + 96*K1*K2*K3**3 + 6736*K1*K2*K3 + 376*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 1672*K2**4 - 32*K2**3*K6 - 816*K2**2*K3**2 - 24*K2**2*K4**2 + 1112*K2**2*K4 - 2188*K2**2 + 224*K2*K3*K5 + 24*K2*K4*K6 - 16*K3**4 - 1684*K3**2 - 214*K4**2 - 4*K5**2 - 4*K6**2 + 2964
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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