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Flat knot 6.285

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,0,1,4,4,3,0,2,1,1,2,2,2,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.285']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.105', '6.144', '6.261', '6.285', '6.392', '6.480']
Outer characteristic polynomial of the knot is: t^7+84t^5+85t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.285']
2-strand cable arrow polynomial of the knot is: -128*K1**2*K2**3*K4 + 1440*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 4608*K1**2*K2**2 + 128*K1**2*K2*K4**2 - 672*K1**2*K2*K4 + 4688*K1**2*K2 - 288*K1**2*K4**2 - 4064*K1**2 + 384*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 608*K1*K2**2*K5 - 960*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5336*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1200*K1*K3*K4 + 816*K1*K4*K5 - 32*K2**4*K4**2 + 320*K2**4*K4 - 1840*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 320*K2**2*K3**2 - 472*K2**2*K4**2 + 2600*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 2990*K2**2 - 32*K2*K3*K4*K5 + 960*K2*K3*K5 + 224*K2*K4*K6 + 40*K2*K5*K7 - 1728*K3**2 - 1082*K4**2 - 432*K5**2 - 18*K6**2 + 3328
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.285']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16363', 'vk6.16406', 'vk6.18119', 'vk6.18454', 'vk6.22693', 'vk6.22794', 'vk6.24572', 'vk6.24988', 'vk6.34658', 'vk6.34745', 'vk6.36709', 'vk6.37130', 'vk6.42319', 'vk6.42366', 'vk6.43981', 'vk6.44295', 'vk6.54624', 'vk6.54651', 'vk6.55935', 'vk6.56228', 'vk6.59104', 'vk6.59166', 'vk6.60469', 'vk6.60829', 'vk6.64645', 'vk6.64693', 'vk6.65589', 'vk6.65895', 'vk6.67998', 'vk6.68024', 'vk6.68662', 'vk6.68872']
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 O1O2O3O4O5U1U5O6U3U2U6U4
R3 orbit {'O1O2O3O4O5U1U5O6U3U2U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U6U4U3O6U1U5
Gauss code of K* O1O2O3O4U5U2U1U4U6O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U5U1U4U3U6
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 -1 3 1 2],[ 4 0 3 2 4 1 2],[ 1 -3 0 0 3 0 2],[ 1 -2 0 0 2 0 1],[-3 -4 -3 -2 0 0 0],[-1 -1 0 0 0 0 0],[-2 -2 -2 -1 0 0 0]]
Primitive based matrix [[ 0 3 2 1 -1 -1 -4],[-3 0 0 0 -2 -3 -4],[-2 0 0 0 -1 -2 -2],[-1 0 0 0 0 0 -1],[ 1 2 1 0 0 0 -2],[ 1 3 2 0 0 0 -3],[ 4 4 2 1 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,1,4,0,0,2,3,4,0,1,2,2,0,0,1,0,2,3]
Phi over symmetry [-4,-1,-1,1,2,3,0,1,4,4,3,0,2,1,1,2,2,2,1,2,1]
Phi of -K [-4,-1,-1,1,2,3,0,1,4,4,3,0,2,1,1,2,2,2,1,2,1]
Phi of K* [-3,-2,-1,1,1,4,1,2,1,2,3,1,1,2,4,2,2,4,0,0,1]
Phi of -K* [-4,-1,-1,1,2,3,2,3,1,2,4,0,0,1,2,0,2,3,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 8z^2+25z+19
Enhanced Jones-Krushkal polynomial -2w^4z^2+10w^3z^2-4w^3z+29w^2z+19w
Inner characteristic polynomial t^6+52t^4+24t^2+1
Outer characteristic polynomial t^7+84t^5+85t^3+13t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + 2
2-strand cable arrow polynomial -128*K1**2*K2**3*K4 + 1440*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 4608*K1**2*K2**2 + 128*K1**2*K2*K4**2 - 672*K1**2*K2*K4 + 4688*K1**2*K2 - 288*K1**2*K4**2 - 4064*K1**2 + 384*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 608*K1*K2**2*K5 - 960*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5336*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1200*K1*K3*K4 + 816*K1*K4*K5 - 32*K2**4*K4**2 + 320*K2**4*K4 - 1840*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 320*K2**2*K3**2 - 472*K2**2*K4**2 + 2600*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 2990*K2**2 - 32*K2*K3*K4*K5 + 960*K2*K3*K5 + 224*K2*K4*K6 + 40*K2*K5*K7 - 1728*K3**2 - 1082*K4**2 - 432*K5**2 - 18*K6**2 + 3328
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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