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Flat knot 6.697

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,2,2,2,0,1,1,1,1,2,2,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.697']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 8*K1*K2 + K1 + 2*K2 + 3*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.697', '6.1075', '6.1524', '6.1733']
Outer characteristic polynomial of the knot is: t^7+40t^5+67t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.697']
2-strand cable arrow polynomial of the knot is: 1248*K1**4*K2 - 3600*K1**4 + 864*K1**3*K2*K3 - 576*K1**3*K3 - 128*K1**2*K2**4 + 416*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5664*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 7888*K1**2*K2 - 1072*K1**2*K3**2 - 32*K1**2*K4**2 - 3872*K1**2 + 320*K1*K2**3*K3 - 1856*K1*K2**2*K3 - 192*K1*K2**2*K5 + 32*K1*K2*K3**3 - 224*K1*K2*K3*K4 + 7272*K1*K2*K3 + 2000*K1*K3*K4 + 104*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 688*K2**4 - 32*K2**3*K6 - 576*K2**2*K3**2 - 64*K2**2*K4**2 + 1544*K2**2*K4 - 3962*K2**2 - 32*K2*K3**2*K4 + 488*K2*K3*K5 + 56*K2*K4*K6 - 32*K3**4 + 48*K3**2*K6 - 2304*K3**2 - 868*K4**2 - 104*K5**2 - 22*K6**2 + 4002
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.697']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4354', 'vk6.4385', 'vk6.5672', 'vk6.5703', 'vk6.7745', 'vk6.7776', 'vk6.9223', 'vk6.9254', 'vk6.10478', 'vk6.10556', 'vk6.10653', 'vk6.10697', 'vk6.10728', 'vk6.10842', 'vk6.14627', 'vk6.15327', 'vk6.15452', 'vk6.16246', 'vk6.17964', 'vk6.24408', 'vk6.30157', 'vk6.30235', 'vk6.30332', 'vk6.30461', 'vk6.33961', 'vk6.34366', 'vk6.34420', 'vk6.43843', 'vk6.50432', 'vk6.50462', 'vk6.54213', 'vk6.63432']
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 O1O2O3O4U3O5U6U4O6U1U5U2
R3 orbit {'O1O2O3O4U3O5U6U4O6U1U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U4O6U1U6O5U2
Gauss code of K* O1O2O3U1U3U4U5O4U2O6O5U6
Gauss code of -K* O1O2O3U4O5O4U2O6U5U6U1U3
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 -2 1 -1 1 2 -1],[ 2 0 2 -1 2 2 1],[-1 -2 0 -1 1 1 -2],[ 1 1 1 0 1 1 0],[-1 -2 -1 -1 0 0 -1],[-2 -2 -1 -1 0 0 -2],[ 1 -1 2 0 1 2 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -1 -1 -2 -2],[-1 0 0 -1 -1 -1 -2],[-1 1 1 0 -1 -2 -2],[ 1 1 1 1 0 0 1],[ 1 2 1 2 0 0 -1],[ 2 2 2 2 -1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,1,1,2,2,1,1,1,2,1,2,2,0,-1,1]
Phi over symmetry [-2,-1,-1,1,1,2,-1,1,2,2,2,0,1,1,1,1,2,2,-1,0,1]
Phi of -K [-2,-1,-1,1,1,2,0,2,1,1,2,0,0,1,1,1,1,2,-1,0,1]
Phi of K* [-2,-1,-1,1,1,2,0,1,1,2,2,1,0,1,1,1,1,1,0,0,2]
Phi of -K* [-2,-1,-1,1,1,2,-1,1,2,2,2,0,1,1,1,1,2,2,-1,0,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+25z+27
Enhanced Jones-Krushkal polynomial 6w^3z^2+25w^2z+27w
Inner characteristic polynomial t^6+28t^4+41t^2+1
Outer characteristic polynomial t^7+40t^5+67t^3+6t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 8*K1*K2 + K1 + 2*K2 + 3*K3 + 3
2-strand cable arrow polynomial 1248*K1**4*K2 - 3600*K1**4 + 864*K1**3*K2*K3 - 576*K1**3*K3 - 128*K1**2*K2**4 + 416*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5664*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 7888*K1**2*K2 - 1072*K1**2*K3**2 - 32*K1**2*K4**2 - 3872*K1**2 + 320*K1*K2**3*K3 - 1856*K1*K2**2*K3 - 192*K1*K2**2*K5 + 32*K1*K2*K3**3 - 224*K1*K2*K3*K4 + 7272*K1*K2*K3 + 2000*K1*K3*K4 + 104*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 688*K2**4 - 32*K2**3*K6 - 576*K2**2*K3**2 - 64*K2**2*K4**2 + 1544*K2**2*K4 - 3962*K2**2 - 32*K2*K3**2*K4 + 488*K2*K3*K5 + 56*K2*K4*K6 - 32*K3**4 + 48*K3**2*K6 - 2304*K3**2 - 868*K4**2 - 104*K5**2 - 22*K6**2 + 4002
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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