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Flat knot 6.24

Min(phi) over symmetries of the knot is: [-5,-2,-1,2,2,4,1,3,2,4,5,1,1,2,3,1,2,3,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.24']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K3 - 8*K1**2 - 10*K1*K2 - 2*K1*K3 - 2*K1*K4 - 2*K1 + 5*K2 + 2*K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.24']
Outer characteristic polynomial of the knot is: t^7+139t^5+71t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.24']
2-strand cable arrow polynomial of the knot is: -752*K1**4 + 192*K1**3*K2*K3 - 288*K1**3*K3 + 448*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 5472*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 6664*K1**2*K2 - 448*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 5104*K1**2 + 2400*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 1472*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 + 64*K1*K2**2*K5*K6 - 512*K1*K2**2*K5 + 32*K1*K2**2*K6*K7 + 96*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 480*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 6912*K1*K2*K3 - 160*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 32*K1*K2*K5*K6 + 32*K1*K3**3*K4 + 1304*K1*K3*K4 + 256*K1*K4*K5 + 48*K1*K5*K6 + 24*K1*K6*K7 - 64*K2**6 - 384*K2**4*K3**2 + 160*K2**4*K4 - 32*K2**4*K6**2 - 2448*K2**4 + 320*K2**3*K3*K5 + 128*K2**3*K4*K6 + 32*K2**3*K6*K8 - 96*K2**3*K6 + 64*K2**2*K3**2*K4 - 2016*K2**2*K3**2 - 32*K2**2*K3*K7 - 488*K2**2*K4**2 - 32*K2**2*K4*K8 + 2440*K2**2*K4 - 160*K2**2*K5**2 - 136*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 3016*K2**2 - 64*K2*K3**2*K4 + 1096*K2*K3*K5 + 432*K2*K4*K6 + 72*K2*K5*K7 + 40*K2*K6*K8 + 8*K2*K7*K9 - 48*K3**4 - 32*K3**2*K4**2 + 40*K3**2*K6 - 2036*K3**2 + 24*K3*K4*K7 - 882*K4**2 - 204*K5**2 - 88*K6**2 - 32*K7**2 - 6*K8**2 + 4062
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.24']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73270', 'vk6.73411', 'vk6.74021', 'vk6.74567', 'vk6.75174', 'vk6.75408', 'vk6.76043', 'vk6.76769', 'vk6.78131', 'vk6.78370', 'vk6.78998', 'vk6.79559', 'vk6.79968', 'vk6.80123', 'vk6.80524', 'vk6.80988', 'vk6.81872', 'vk6.82153', 'vk6.82180', 'vk6.82588', 'vk6.83577', 'vk6.83756', 'vk6.84036', 'vk6.84602', 'vk6.84930', 'vk6.85582', 'vk6.85704', 'vk6.85928', 'vk6.86727', 'vk6.87662', 'vk6.88929', 'vk6.89977']
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 O1O2O3O4O5O6U1U3U6U2U4U5
R3 orbit {'O1O2O3O4O5O6U1U3U6U2U4U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U2U3U5U1U4U6
Gauss code of K* O1O2O3O4O5O6U1U4U2U5U6U3
Gauss code of -K* O1O2O3O4O5O6U4U1U2U5U3U6
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 -5 -1 -2 2 4 2],[ 5 0 3 1 4 5 2],[ 1 -3 0 -1 2 3 1],[ 2 -1 1 0 2 3 1],[-2 -4 -2 -2 0 1 0],[-4 -5 -3 -3 -1 0 0],[-2 -2 -1 -1 0 0 0]]
Primitive based matrix [[ 0 4 2 2 -1 -2 -5],[-4 0 0 -1 -3 -3 -5],[-2 0 0 0 -1 -1 -2],[-2 1 0 0 -2 -2 -4],[ 1 3 1 2 0 -1 -3],[ 2 3 1 2 1 0 -1],[ 5 5 2 4 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-2,1,2,5,0,1,3,3,5,0,1,1,2,2,2,4,1,3,1]
Phi over symmetry [-5,-2,-1,2,2,4,1,3,2,4,5,1,1,2,3,1,2,3,0,0,1]
Phi of -K [-5,-2,-1,2,2,4,2,1,3,5,4,0,2,3,3,1,2,2,0,1,2]
Phi of K* [-4,-2,-2,1,2,5,1,2,2,3,4,0,1,2,3,2,3,5,0,1,2]
Phi of -K* [-5,-2,-1,2,2,4,1,3,2,4,5,1,1,2,3,1,2,3,0,0,1]
Symmetry type of based matrix c
u-polynomial t^5-t^4-t^2+t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2+21w^2z+27w
Inner characteristic polynomial t^6+85t^4+12t^2
Outer characteristic polynomial t^7+139t^5+71t^3+4t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K3 - 8*K1**2 - 10*K1*K2 - 2*K1*K3 - 2*K1*K4 - 2*K1 + 5*K2 + 2*K3 + K4 + 5
2-strand cable arrow polynomial -752*K1**4 + 192*K1**3*K2*K3 - 288*K1**3*K3 + 448*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 5472*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 6664*K1**2*K2 - 448*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 5104*K1**2 + 2400*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 1472*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 + 64*K1*K2**2*K5*K6 - 512*K1*K2**2*K5 + 32*K1*K2**2*K6*K7 + 96*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 480*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 6912*K1*K2*K3 - 160*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 32*K1*K2*K5*K6 + 32*K1*K3**3*K4 + 1304*K1*K3*K4 + 256*K1*K4*K5 + 48*K1*K5*K6 + 24*K1*K6*K7 - 64*K2**6 - 384*K2**4*K3**2 + 160*K2**4*K4 - 32*K2**4*K6**2 - 2448*K2**4 + 320*K2**3*K3*K5 + 128*K2**3*K4*K6 + 32*K2**3*K6*K8 - 96*K2**3*K6 + 64*K2**2*K3**2*K4 - 2016*K2**2*K3**2 - 32*K2**2*K3*K7 - 488*K2**2*K4**2 - 32*K2**2*K4*K8 + 2440*K2**2*K4 - 160*K2**2*K5**2 - 136*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 3016*K2**2 - 64*K2*K3**2*K4 + 1096*K2*K3*K5 + 432*K2*K4*K6 + 72*K2*K5*K7 + 40*K2*K6*K8 + 8*K2*K7*K9 - 48*K3**4 - 32*K3**2*K4**2 + 40*K3**2*K6 - 2036*K3**2 + 24*K3*K4*K7 - 882*K4**2 - 204*K5**2 - 88*K6**2 - 32*K7**2 - 6*K8**2 + 4062
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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