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Flat knot 6.35

Min(phi) over symmetries of the knot is: [-5,-1,0,1,1,4,1,3,2,5,4,1,1,2,2,1,1,3,0,1,3]
Flat knots (up to 7 crossings) with same phi are :['6.35']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 + 4*K1**2*K3 - 10*K1**2 - 10*K1*K2 - 2*K1*K3 - 2*K1*K4 - 2*K1 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.35']
Outer characteristic polynomial of the knot is: t^7+130t^5+100t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.35']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 384*K1**4*K2**2 + 1728*K1**4*K2 - 2992*K1**4 - 640*K1**3*K2**2*K3 + 1344*K1**3*K2*K3 - 576*K1**3*K3 - 256*K1**2*K2**4 + 256*K1**2*K2**3*K3**2 + 2944*K1**2*K2**3 - 640*K1**2*K2**2*K3**2 + 320*K1**2*K2**2*K4 - 10672*K1**2*K2**2 + 320*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 1088*K1**2*K2*K4 + 10352*K1**2*K2 - 496*K1**2*K3**2 - 128*K1**2*K4**2 - 5960*K1**2 - 256*K1*K2**4*K3 - 256*K1*K2**3*K3*K4 + 3136*K1*K2**3*K3 + 864*K1*K2**2*K3*K4 - 2880*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 + 32*K1*K2**2*K5*K6 - 704*K1*K2**2*K5 + 64*K1*K2*K3**3 - 576*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 9624*K1*K2*K3 - 64*K1*K2*K4*K5 + 1528*K1*K3*K4 + 272*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**6 - 448*K2**4*K3**2 - 32*K2**4*K4**2 + 672*K2**4*K4 - 32*K2**4*K6**2 - 3944*K2**4 + 640*K2**3*K3*K5 + 160*K2**3*K4*K6 + 32*K2**3*K5*K7 + 32*K2**3*K6*K8 - 160*K2**3*K6 + 64*K2**2*K3**2*K6 - 2560*K2**2*K3**2 + 32*K2**2*K3*K4*K7 - 96*K2**2*K3*K7 - 904*K2**2*K4**2 - 32*K2**2*K4*K8 + 3608*K2**2*K4 - 304*K2**2*K5**2 - 136*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 3952*K2**2 - 128*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1568*K2*K3*K5 + 432*K2*K4*K6 + 112*K2*K5*K7 + 32*K2*K6*K8 + 40*K3**2*K6 - 2672*K3**2 + 16*K3*K4*K7 - 1120*K4**2 - 368*K5**2 - 80*K6**2 - 16*K7**2 - 4*K8**2 + 5530
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.35']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19932', 'vk6.19990', 'vk6.21167', 'vk6.21254', 'vk6.26881', 'vk6.27017', 'vk6.28641', 'vk6.28735', 'vk6.38308', 'vk6.38421', 'vk6.40437', 'vk6.40599', 'vk6.45174', 'vk6.45305', 'vk6.47010', 'vk6.47083', 'vk6.56718', 'vk6.56796', 'vk6.57807', 'vk6.57927', 'vk6.61134', 'vk6.61296', 'vk6.62381', 'vk6.62483', 'vk6.66411', 'vk6.66500', 'vk6.67173', 'vk6.67289', 'vk6.69060', 'vk6.69150', 'vk6.69847', 'vk6.69907']
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 O1O2O3O4O5O6U1U4U5U3U6U2
R3 orbit {'O1O2O3O4O5O6U1U4U5U3U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U1U4U2U3U6
Gauss code of K* O1O2O3O4O5O6U1U6U4U2U3U5
Gauss code of -K* O1O2O3O4O5O6U2U4U5U3U1U6
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 -5 1 0 -1 1 4],[ 5 0 5 3 1 2 4],[-1 -5 0 -1 -2 0 3],[ 0 -3 1 0 -1 1 3],[ 1 -1 2 1 0 1 2],[-1 -2 0 -1 -1 0 1],[-4 -4 -3 -3 -2 -1 0]]
Primitive based matrix [[ 0 4 1 1 0 -1 -5],[-4 0 -1 -3 -3 -2 -4],[-1 1 0 0 -1 -1 -2],[-1 3 0 0 -1 -2 -5],[ 0 3 1 1 0 -1 -3],[ 1 2 1 2 1 0 -1],[ 5 4 2 5 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,-1,0,1,5,1,3,3,2,4,0,1,1,2,1,2,5,1,3,1]
Phi over symmetry [-5,-1,0,1,1,4,1,3,2,5,4,1,1,2,2,1,1,3,0,1,3]
Phi of -K [-5,-1,0,1,1,4,3,2,1,4,5,0,0,1,3,0,0,1,0,0,2]
Phi of K* [-4,-1,-1,0,1,5,0,2,1,3,5,0,0,0,1,0,1,4,0,2,3]
Phi of -K* [-5,-1,0,1,1,4,1,3,2,5,4,1,1,2,2,1,1,3,0,1,3]
Symmetry type of based matrix c
u-polynomial t^5-t^4-t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+86t^4+26t^2+1
Outer characteristic polynomial t^7+130t^5+100t^3+11t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 + 4*K1**2*K3 - 10*K1**2 - 10*K1*K2 - 2*K1*K3 - 2*K1*K4 - 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -128*K1**6 - 384*K1**4*K2**2 + 1728*K1**4*K2 - 2992*K1**4 - 640*K1**3*K2**2*K3 + 1344*K1**3*K2*K3 - 576*K1**3*K3 - 256*K1**2*K2**4 + 256*K1**2*K2**3*K3**2 + 2944*K1**2*K2**3 - 640*K1**2*K2**2*K3**2 + 320*K1**2*K2**2*K4 - 10672*K1**2*K2**2 + 320*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 1088*K1**2*K2*K4 + 10352*K1**2*K2 - 496*K1**2*K3**2 - 128*K1**2*K4**2 - 5960*K1**2 - 256*K1*K2**4*K3 - 256*K1*K2**3*K3*K4 + 3136*K1*K2**3*K3 + 864*K1*K2**2*K3*K4 - 2880*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 + 32*K1*K2**2*K5*K6 - 704*K1*K2**2*K5 + 64*K1*K2*K3**3 - 576*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 9624*K1*K2*K3 - 64*K1*K2*K4*K5 + 1528*K1*K3*K4 + 272*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**6 - 448*K2**4*K3**2 - 32*K2**4*K4**2 + 672*K2**4*K4 - 32*K2**4*K6**2 - 3944*K2**4 + 640*K2**3*K3*K5 + 160*K2**3*K4*K6 + 32*K2**3*K5*K7 + 32*K2**3*K6*K8 - 160*K2**3*K6 + 64*K2**2*K3**2*K6 - 2560*K2**2*K3**2 + 32*K2**2*K3*K4*K7 - 96*K2**2*K3*K7 - 904*K2**2*K4**2 - 32*K2**2*K4*K8 + 3608*K2**2*K4 - 304*K2**2*K5**2 - 136*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 3952*K2**2 - 128*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1568*K2*K3*K5 + 432*K2*K4*K6 + 112*K2*K5*K7 + 32*K2*K6*K8 + 40*K3**2*K6 - 2672*K3**2 + 16*K3*K4*K7 - 1120*K4**2 - 368*K5**2 - 80*K6**2 - 16*K7**2 - 4*K8**2 + 5530
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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