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Flat knot 6.89

Min(phi) over symmetries of the knot is: [-4,-2,0,0,2,4,0,2,3,2,5,1,1,1,2,0,1,2,1,3,0]
Flat knots (up to 7 crossings) with same phi are :['6.89']
Arrow polynomial of the knot is: 8*K1**2*K2 - 8*K1**2 - 4*K2**2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.89', '6.1214', '6.1929']
Outer characteristic polynomial of the knot is: t^7+112t^5+59t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.89']
2-strand cable arrow polynomial of the knot is: -64*K2**4*K4**2 + 256*K2**4*K4 - 1088*K2**4 + 64*K2**2*K4**3 - 480*K2**2*K4**2 + 1632*K2**2*K4 - 568*K2**2 + 304*K2*K4*K6 + 16*K2*K6*K8 - 16*K4**4 - 568*K4**2 - 72*K6**2 - 8*K8**2 + 590
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.89']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72577', 'vk6.72686', 'vk6.73000', 'vk6.73152', 'vk6.73216', 'vk6.73238', 'vk6.73248', 'vk6.74923', 'vk6.75129', 'vk6.75135', 'vk6.75152', 'vk6.75344', 'vk6.76487', 'vk6.77861', 'vk6.77899', 'vk6.78073', 'vk6.78097', 'vk6.78098', 'vk6.79346', 'vk6.80801', 'vk6.85073', 'vk6.85606', 'vk6.85765', 'vk6.87315', 'vk6.90163', 'vk6.90192']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4O5O6U2U5U4U1U6U3
R3 orbit {'O1O2O3O4O5U1U4U6U2U5O6U3', 'O1O2O3O4O5O6U2U5U4U1U6U3'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5O6U4U1U6U3U2U5
Gauss code of K* O1O2O3O4O5O6U4U1U6U3U2U5
Gauss code of -K* Same
Diagrammatic symmetry type -
Flat genus of the diagram 2
If K is checkerboard colorable True
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -4 2 0 0 4],[ 2 0 -2 3 1 1 4],[ 4 2 0 4 2 1 3],[-2 -3 -4 0 -1 -1 2],[ 0 -1 -2 1 0 0 2],[ 0 -1 -1 1 0 0 1],[-4 -4 -3 -2 -2 -1 0]]
Primitive based matrix [[ 0 4 2 0 0 -2 -4],[-4 0 -2 -1 -2 -4 -3],[-2 2 0 -1 -1 -3 -4],[ 0 1 1 0 0 -1 -1],[ 0 2 1 0 0 -1 -2],[ 2 4 3 1 1 0 -2],[ 4 3 4 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,0,0,2,4,2,1,2,4,3,1,1,3,4,0,1,1,1,2,2]
Phi over symmetry [-4,-2,0,0,2,4,0,2,3,2,5,1,1,1,2,0,1,2,1,3,0]
Phi of -K [-4,-2,0,0,2,4,0,2,3,2,5,1,1,1,2,0,1,2,1,3,0]
Phi of K* [-4,-2,0,0,2,4,0,2,3,2,5,1,1,1,2,0,1,2,1,3,0]
Phi of -K* [-4,-2,0,0,2,4,2,1,2,4,3,1,1,3,4,0,1,1,1,2,2]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+17z+11
Enhanced Jones-Krushkal polynomial 6w^3z^2+17w^2z+11
Inner characteristic polynomial t^6+72t^4+35t^2
Outer characteristic polynomial t^7+112t^5+59t^3
Flat arrow polynomial 8*K1**2*K2 - 8*K1**2 - 4*K2**2 + 5
2-strand cable arrow polynomial -64*K2**4*K4**2 + 256*K2**4*K4 - 1088*K2**4 + 64*K2**2*K4**3 - 480*K2**2*K4**2 + 1632*K2**2*K4 - 568*K2**2 + 304*K2*K4*K6 + 16*K2*K6*K8 - 16*K4**4 - 568*K4**2 - 72*K6**2 - 8*K8**2 + 590
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}]]
If K is slice True
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