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Flat knot 6.1160

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,3,1,2,1,0,0,0,1,0,0,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1160']
Arrow polynomial of the knot is: -2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K2 + K3 + 2*K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.373', '6.434', '6.878', '6.886', '6.952', '6.1160']
Outer characteristic polynomial of the knot is: t^7+36t^5+95t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1160']
2-strand cable arrow polynomial of the knot is: -1392*K1**4 + 384*K1**3*K2*K3 + 128*K1**3*K3*K4 - 320*K1**3*K3 - 1056*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 4016*K1**2*K2 - 1840*K1**2*K3**2 - 352*K1**2*K3*K5 - 576*K1**2*K4**2 - 4592*K1**2 - 864*K1*K2**2*K3 - 64*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5504*K1*K2*K3 - 64*K1*K2*K4*K7 - 32*K1*K2*K5*K6 + 3968*K1*K3*K4 + 808*K1*K4*K5 + 72*K1*K5*K6 + 48*K1*K6*K7 - 32*K2**4 - 32*K2**3*K6 - 272*K2**2*K3**2 - 72*K2**2*K4**2 + 1112*K2**2*K4 - 8*K2**2*K6**2 - 3778*K2**2 + 480*K2*K3*K5 + 264*K2*K4*K6 + 96*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 - 48*K3**2*K4**2 + 120*K3**2*K6 - 3008*K3**2 + 112*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1778*K4**2 - 396*K5**2 - 174*K6**2 - 92*K7**2 - 12*K8**2 + 4484
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1160']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3627', 'vk6.3712', 'vk6.3905', 'vk6.4012', 'vk6.7053', 'vk6.7108', 'vk6.7285', 'vk6.7386', 'vk6.11391', 'vk6.12576', 'vk6.12689', 'vk6.19109', 'vk6.19154', 'vk6.19814', 'vk6.25722', 'vk6.25781', 'vk6.26253', 'vk6.26696', 'vk6.30989', 'vk6.31118', 'vk6.32173', 'vk6.37837', 'vk6.37892', 'vk6.44982', 'vk6.48255', 'vk6.48436', 'vk6.50015', 'vk6.50160', 'vk6.52148', 'vk6.63722', 'vk6.66210', 'vk6.66237']
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 O1O2O3O4U1U4U3O5O6U5U2U6
R3 orbit {'O1O2O3O4U1U4U3O5O6U5U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3U6O5O6U2U1U4
Gauss code of K* O1O2O3U4U5O4O6O5U1U6U3U2
Gauss code of -K* O1O2O3U1U3O4O5O6U5U4U2U6
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 0 1 1 -1 2],[ 3 0 3 2 1 0 1],[ 0 -3 0 0 0 0 2],[-1 -2 0 0 0 0 0],[-1 -1 0 0 0 0 0],[ 1 0 0 0 0 0 1],[-2 -1 -2 0 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 0 -2 -1 -1],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 0 -2],[ 0 2 0 0 0 0 -3],[ 1 1 0 0 0 0 0],[ 3 1 1 2 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,0,2,1,1,0,0,0,1,0,0,2,0,3,0]
Phi over symmetry [-3,-1,0,1,1,2,0,3,1,2,1,0,0,0,1,0,0,2,0,0,0]
Phi of -K [-3,-1,0,1,1,2,2,0,2,3,4,1,2,2,2,1,1,0,0,1,1]
Phi of K* [-2,-1,-1,0,1,3,1,1,0,2,4,0,1,2,2,1,2,3,1,0,2]
Phi of -K* [-3,-1,0,1,1,2,0,3,1,2,1,0,0,0,1,0,0,2,0,0,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+20t^4+34t^2
Outer characteristic polynomial t^7+36t^5+95t^3+7t
Flat arrow polynomial -2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K2 + K3 + 2*K4 + 2
2-strand cable arrow polynomial -1392*K1**4 + 384*K1**3*K2*K3 + 128*K1**3*K3*K4 - 320*K1**3*K3 - 1056*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 4016*K1**2*K2 - 1840*K1**2*K3**2 - 352*K1**2*K3*K5 - 576*K1**2*K4**2 - 4592*K1**2 - 864*K1*K2**2*K3 - 64*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5504*K1*K2*K3 - 64*K1*K2*K4*K7 - 32*K1*K2*K5*K6 + 3968*K1*K3*K4 + 808*K1*K4*K5 + 72*K1*K5*K6 + 48*K1*K6*K7 - 32*K2**4 - 32*K2**3*K6 - 272*K2**2*K3**2 - 72*K2**2*K4**2 + 1112*K2**2*K4 - 8*K2**2*K6**2 - 3778*K2**2 + 480*K2*K3*K5 + 264*K2*K4*K6 + 96*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 - 48*K3**2*K4**2 + 120*K3**2*K6 - 3008*K3**2 + 112*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1778*K4**2 - 396*K5**2 - 174*K6**2 - 92*K7**2 - 12*K8**2 + 4484
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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