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Flat knot 6.496

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,0,2,4,4,3,1,1,2,1,1,1,2,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.496']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 14*K1**2 - 6*K1*K2 - 2*K1*K3 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.273', '6.496']
Outer characteristic polynomial of the knot is: t^7+70t^5+75t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.496']
2-strand cable arrow polynomial of the knot is: -256*K1**6 + 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 2944*K1**4*K2 - 4368*K1**4 - 128*K1**3*K2**2*K3 + 736*K1**3*K2*K3 - 800*K1**3*K3 - 384*K1**2*K2**4 + 2720*K1**2*K2**3 - 9840*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 768*K1**2*K2*K4 + 11552*K1**2*K2 - 336*K1**2*K3**2 - 128*K1**2*K4**2 - 5984*K1**2 + 864*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 2752*K1*K2**2*K3 - 320*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 9528*K1*K2*K3 - 64*K1*K2*K4*K5 + 1352*K1*K3*K4 + 280*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 2488*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 128*K2**3*K6 - 1088*K2**2*K3**2 - 32*K2**2*K3*K7 - 312*K2**2*K4**2 + 3304*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 5480*K2**2 + 1112*K2*K3*K5 + 288*K2*K4*K6 + 16*K2*K5*K7 - 2584*K3**2 - 1092*K4**2 - 312*K5**2 - 64*K6**2 + 5786
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.496']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11050', 'vk6.11130', 'vk6.12216', 'vk6.12325', 'vk6.16420', 'vk6.19238', 'vk6.19317', 'vk6.19531', 'vk6.19610', 'vk6.22720', 'vk6.22821', 'vk6.26046', 'vk6.26085', 'vk6.26427', 'vk6.26507', 'vk6.30627', 'vk6.30724', 'vk6.31935', 'vk6.34767', 'vk6.38119', 'vk6.38123', 'vk6.42381', 'vk6.44641', 'vk6.44747', 'vk6.51843', 'vk6.52709', 'vk6.52809', 'vk6.56582', 'vk6.56627', 'vk6.64713', 'vk6.66283', 'vk6.66299']
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 O1O2O3O4O5U1U5U3O6U2U6U4
R3 orbit {'O1O2O3O4O5U1U5U3O6U2U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U6U4O6U3U1U5
Gauss code of K* O1O2O3U4O5O4O6U1U5U3U6U2
Gauss code of -K* O1O2O3U2O4O5O6U5U1U4U3U6
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 -4 -1 0 3 1 1],[ 4 0 3 2 4 1 1],[ 1 -3 0 0 3 0 1],[ 0 -2 0 0 1 0 0],[-3 -4 -3 -1 0 0 0],[-1 -1 0 0 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 3 1 1 0 -1 -4],[-3 0 0 0 -1 -3 -4],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 -1 -1],[ 0 1 0 0 0 0 -2],[ 1 3 0 1 0 0 -3],[ 4 4 1 1 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,0,1,4,0,0,1,3,4,0,0,0,1,0,1,1,0,2,3]
Phi over symmetry [-4,-1,0,1,1,3,0,2,4,4,3,1,1,2,1,1,1,2,0,2,2]
Phi of -K [-4,-1,0,1,1,3,0,2,4,4,3,1,1,2,1,1,1,2,0,2,2]
Phi of K* [-3,-1,-1,0,1,4,2,2,2,1,3,0,1,1,4,1,2,4,1,2,0]
Phi of -K* [-4,-1,0,1,1,3,3,2,1,1,4,0,0,1,3,0,0,1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 4z^2+25z+35
Enhanced Jones-Krushkal polynomial 4w^3z^2+25w^2z+35w
Inner characteristic polynomial t^6+42t^4+27t^2+1
Outer characteristic polynomial t^7+70t^5+75t^3+11t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 14*K1**2 - 6*K1*K2 - 2*K1*K3 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -256*K1**6 + 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 2944*K1**4*K2 - 4368*K1**4 - 128*K1**3*K2**2*K3 + 736*K1**3*K2*K3 - 800*K1**3*K3 - 384*K1**2*K2**4 + 2720*K1**2*K2**3 - 9840*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 768*K1**2*K2*K4 + 11552*K1**2*K2 - 336*K1**2*K3**2 - 128*K1**2*K4**2 - 5984*K1**2 + 864*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 2752*K1*K2**2*K3 - 320*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 9528*K1*K2*K3 - 64*K1*K2*K4*K5 + 1352*K1*K3*K4 + 280*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 2488*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 128*K2**3*K6 - 1088*K2**2*K3**2 - 32*K2**2*K3*K7 - 312*K2**2*K4**2 + 3304*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 5480*K2**2 + 1112*K2*K3*K5 + 288*K2*K4*K6 + 16*K2*K5*K7 - 2584*K3**2 - 1092*K4**2 - 312*K5**2 - 64*K6**2 + 5786
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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