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Flat knot 6.1357

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,0,1,1,2,4,0,1,0,1,0,0,0,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1357']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1*K2 - 2*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.233', '6.340', '6.382', '6.550', '6.656', '6.663', '6.683', '6.698', '6.739', '6.745', '6.759', '6.765', '6.1357', '6.1358', '6.1370']
Outer characteristic polynomial of the knot is: t^7+39t^5+56t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1357']
2-strand cable arrow polynomial of the knot is: 1248*K1**4*K2 - 3040*K1**4 + 672*K1**3*K2*K3 - 608*K1**3*K3 - 128*K1**2*K2**4 + 480*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4128*K1**2*K2**2 - 224*K1**2*K2*K4 + 6128*K1**2*K2 - 736*K1**2*K3**2 - 32*K1**2*K4**2 - 2800*K1**2 + 224*K1*K2**3*K3 - 864*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 4344*K1*K2*K3 + 664*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 432*K2**4 - 192*K2**2*K3**2 - 8*K2**2*K4**2 + 368*K2**2*K4 - 2368*K2**2 + 56*K2*K3*K5 - 1144*K3**2 - 116*K4**2 + 2554
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1357']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10528', 'vk6.10532', 'vk6.10617', 'vk6.10625', 'vk6.10806', 'vk6.10814', 'vk6.10905', 'vk6.10909', 'vk6.19019', 'vk6.19035', 'vk6.19084', 'vk6.19086', 'vk6.19131', 'vk6.19133', 'vk6.25535', 'vk6.25551', 'vk6.25632', 'vk6.25648', 'vk6.25756', 'vk6.25758', 'vk6.30213', 'vk6.30217', 'vk6.30304', 'vk6.30312', 'vk6.30433', 'vk6.30441', 'vk6.37729', 'vk6.37745', 'vk6.56508', 'vk6.56516', 'vk6.66172', 'vk6.66180']
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 O1O2O3U2O4O5U1U5U4O6U3U6
R3 orbit {'O1O2O3U2O4O5U1U5U4O6U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U1O4U5U6U3O6O5U2
Gauss code of K* O1O2O3U4O5O4U1U6U5O6U3U2
Gauss code of -K* O1O2O3U2U1O4U5U4U3O6O5U6
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 1 1 1 1],[ 3 0 0 4 2 1 1],[ 1 0 0 1 0 0 1],[-1 -4 -1 0 0 0 1],[-1 -2 0 0 0 0 0],[-1 -1 0 0 0 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 1 0 0 -1 -4],[-1 -1 0 0 0 -1 -1],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 0 -2],[ 1 1 1 0 0 0 0],[ 3 4 1 1 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,1,3,-1,0,0,1,4,0,0,1,1,0,0,1,0,2,0]
Phi over symmetry [-3,-1,1,1,1,1,0,1,1,2,4,0,1,0,1,0,0,0,0,-1,0]
Phi of -K [-3,-1,1,1,1,1,2,0,2,3,3,1,2,1,2,0,-1,0,0,0,0]
Phi of K* [-1,-1,-1,-1,1,3,-1,0,0,1,3,0,0,1,0,0,2,2,2,3,2]
Phi of -K* [-3,-1,1,1,1,1,0,1,1,2,4,0,1,0,1,0,0,0,0,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial 6w^3z^2-4w^3z+27w^2z+23w
Inner characteristic polynomial t^6+25t^4+24t^2
Outer characteristic polynomial t^7+39t^5+56t^3+8t
Flat arrow polynomial 4*K1**3 - 2*K1*K2 - 2*K1 + 1
2-strand cable arrow polynomial 1248*K1**4*K2 - 3040*K1**4 + 672*K1**3*K2*K3 - 608*K1**3*K3 - 128*K1**2*K2**4 + 480*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4128*K1**2*K2**2 - 224*K1**2*K2*K4 + 6128*K1**2*K2 - 736*K1**2*K3**2 - 32*K1**2*K4**2 - 2800*K1**2 + 224*K1*K2**3*K3 - 864*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 4344*K1*K2*K3 + 664*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 432*K2**4 - 192*K2**2*K3**2 - 8*K2**2*K4**2 + 368*K2**2*K4 - 2368*K2**2 + 56*K2*K3*K5 - 1144*K3**2 - 116*K4**2 + 2554
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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