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Flat knot 6.234

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,0,1,4,2,4,0,1,0,1,2,1,3,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.234']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 2*K1*K3 + 5*K2 + 2*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.234', '6.339']
Outer characteristic polynomial of the knot is: t^7+90t^5+97t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.234']
2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 1216*K1**4*K2 - 3616*K1**4 + 832*K1**3*K2*K3 + 32*K1**3*K3*K4 - 512*K1**3*K3 - 256*K1**2*K2**4 + 1344*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 192*K1**2*K2**2*K4 - 8688*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 928*K1**2*K2*K4 + 10280*K1**2*K2 - 1216*K1**2*K3**2 - 32*K1**2*K3*K5 - 160*K1**2*K4**2 - 5644*K1**2 + 1280*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 - 608*K1*K2**2*K5 + 64*K1*K2*K3**3 - 416*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10256*K1*K2*K3 - 32*K1*K2*K4*K5 + 1904*K1*K3*K4 + 192*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 288*K2**4*K4 - 2576*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1776*K2**2*K3**2 - 440*K2**2*K4**2 + 2816*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 4704*K2**2 - 64*K2*K3**2*K4 + 1376*K2*K3*K5 + 280*K2*K4*K6 + 8*K2*K5*K7 - 64*K3**4 - 32*K3**2*K4**2 + 96*K3**2*K6 - 2880*K3**2 + 32*K3*K4*K7 - 962*K4**2 - 236*K5**2 - 72*K6**2 - 8*K7**2 + 5472
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.234']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16536', 'vk6.16627', 'vk6.17515', 'vk6.17571', 'vk6.18873', 'vk6.18951', 'vk6.19216', 'vk6.19509', 'vk6.23060', 'vk6.24111', 'vk6.25499', 'vk6.25574', 'vk6.26024', 'vk6.26408', 'vk6.34931', 'vk6.35045', 'vk6.36296', 'vk6.36362', 'vk6.37600', 'vk6.37689', 'vk6.42503', 'vk6.42613', 'vk6.43473', 'vk6.44605', 'vk6.54765', 'vk6.54858', 'vk6.56440', 'vk6.56562', 'vk6.59287', 'vk6.60187', 'vk6.66097', 'vk6.66138']
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 O1O2O3O4O5U3O6U2U6U4U1U5
R3 orbit {'O1O2O3O4O5U3O6U2U6U4U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U5U2U6U4O6U3
Gauss code of K* O1O2O3O4O5U4U1U6U3U5O6U2
Gauss code of -K* O1O2O3O4O5U4O6U1U3U6U5U2
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 -1 -3 -2 1 4 1],[ 1 0 -2 -1 2 4 1],[ 3 2 0 0 3 4 1],[ 2 1 0 0 1 2 0],[-1 -2 -3 -1 0 1 0],[-4 -4 -4 -2 -1 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 4 1 1 -1 -2 -3],[-4 0 0 -1 -4 -2 -4],[-1 0 0 0 -1 0 -1],[-1 1 0 0 -2 -1 -3],[ 1 4 1 2 0 -1 -2],[ 2 2 0 1 1 0 0],[ 3 4 1 3 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,-1,1,2,3,0,1,4,2,4,0,1,0,1,2,1,3,1,2,0]
Phi over symmetry [-4,-1,-1,1,2,3,0,1,4,2,4,0,1,0,1,2,1,3,1,2,0]
Phi of -K [-3,-2,-1,1,1,4,1,0,1,3,3,0,2,3,4,0,1,1,0,2,3]
Phi of K* [-4,-1,-1,1,2,3,2,3,1,4,3,0,0,2,1,1,3,3,0,0,1]
Phi of -K* [-3,-2,-1,1,1,4,0,2,1,3,4,1,0,1,2,1,2,4,0,0,1]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t^2-t
Normalized Jones-Krushkal polynomial 4z^2+25z+35
Enhanced Jones-Krushkal polynomial 4w^3z^2+25w^2z+35w
Inner characteristic polynomial t^6+58t^4+24t^2+1
Outer characteristic polynomial t^7+90t^5+97t^3+7t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 2*K1*K3 + 5*K2 + 2*K3 + 6
2-strand cable arrow polynomial -320*K1**4*K2**2 + 1216*K1**4*K2 - 3616*K1**4 + 832*K1**3*K2*K3 + 32*K1**3*K3*K4 - 512*K1**3*K3 - 256*K1**2*K2**4 + 1344*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 192*K1**2*K2**2*K4 - 8688*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 928*K1**2*K2*K4 + 10280*K1**2*K2 - 1216*K1**2*K3**2 - 32*K1**2*K3*K5 - 160*K1**2*K4**2 - 5644*K1**2 + 1280*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 - 608*K1*K2**2*K5 + 64*K1*K2*K3**3 - 416*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10256*K1*K2*K3 - 32*K1*K2*K4*K5 + 1904*K1*K3*K4 + 192*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 288*K2**4*K4 - 2576*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1776*K2**2*K3**2 - 440*K2**2*K4**2 + 2816*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 4704*K2**2 - 64*K2*K3**2*K4 + 1376*K2*K3*K5 + 280*K2*K4*K6 + 8*K2*K5*K7 - 64*K3**4 - 32*K3**2*K4**2 + 96*K3**2*K6 - 2880*K3**2 + 32*K3*K4*K7 - 962*K4**2 - 236*K5**2 - 72*K6**2 - 8*K7**2 + 5472
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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