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Flat knot 6.373

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,1,2,3,1,0,1,1,1,1,1,2,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.373']
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+48t^5+75t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.373']
2-strand cable arrow polynomial of the knot is: 640*K1**4*K2 - 2768*K1**4 + 448*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1088*K1**3*K3 + 96*K1**2*K2**2*K4 - 2528*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 160*K1**2*K2*K4 + 7616*K1**2*K2 - 1552*K1**2*K3**2 - 96*K1**2*K3*K5 - 224*K1**2*K4**2 - 64*K1**2*K5**2 - 6400*K1**2 - 864*K1*K2**2*K3 - 32*K1*K2**2*K5 + 32*K1*K2*K3**3 - 608*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 7192*K1*K2*K3 - 32*K1*K3**2*K5 + 2792*K1*K3*K4 + 752*K1*K4*K5 + 192*K1*K5*K6 - 64*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 40*K2**2*K4**2 + 1088*K2**2*K4 - 8*K2**2*K6**2 - 5346*K2**2 - 64*K2*K3**2*K4 + 912*K2*K3*K5 + 192*K2*K4*K6 + 8*K2*K5*K7 + 16*K2*K6*K8 - 64*K3**4 - 16*K3**2*K4**2 + 144*K3**2*K6 - 3392*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1458*K4**2 - 580*K5**2 - 174*K6**2 - 12*K7**2 - 12*K8**2 + 5828
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.373']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4869', 'vk6.5214', 'vk6.6451', 'vk6.6872', 'vk6.8412', 'vk6.8833', 'vk6.9764', 'vk6.10057', 'vk6.11689', 'vk6.12040', 'vk6.13031', 'vk6.20499', 'vk6.20762', 'vk6.21864', 'vk6.27907', 'vk6.29405', 'vk6.29725', 'vk6.32674', 'vk6.33015', 'vk6.39336', 'vk6.39794', 'vk6.46354', 'vk6.47604', 'vk6.47929', 'vk6.48827', 'vk6.49098', 'vk6.51346', 'vk6.51559', 'vk6.53284', 'vk6.57368', 'vk6.64345', 'vk6.66921']
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 O1O2O3O4O5U3U5O6U4U2U1U6
R3 orbit {'O1O2O3O4O5U3U5O6U4U2U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U5U4U2O6U1U3
Gauss code of K* O1O2O3O4U3U2U5U1U6O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U5U4U6U3U2
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 -1 -2 0 1 3],[ 1 0 0 -2 1 1 3],[ 1 0 0 -2 1 1 2],[ 2 2 2 0 2 1 1],[ 0 -1 -1 -2 0 0 1],[-1 -1 -1 -1 0 0 0],[-3 -3 -2 -1 -1 0 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 0 -1 -2 -3 -1],[-1 0 0 0 -1 -1 -1],[ 0 1 0 0 -1 -1 -2],[ 1 2 1 1 0 0 -2],[ 1 3 1 1 0 0 -2],[ 2 1 1 2 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,0,1,2,3,1,0,1,1,1,1,1,2,0,2,2]
Phi over symmetry [-3,-1,0,1,1,2,0,1,2,3,1,0,1,1,1,1,1,2,0,2,2]
Phi of -K [-2,-1,-1,0,1,3,-1,-1,0,2,4,0,0,1,1,0,1,2,1,2,2]
Phi of K* [-3,-1,0,1,1,2,2,2,1,2,4,1,1,1,2,0,0,0,0,-1,-1]
Phi of -K* [-2,-1,-1,0,1,3,2,2,2,1,1,0,1,1,2,1,1,3,0,1,0]
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+32t^4+26t^2+1
Outer characteristic polynomial t^7+48t^5+75t^3+5t
Flat arrow polynomial -2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K2 + K3 + 2*K4 + 2
2-strand cable arrow polynomial 640*K1**4*K2 - 2768*K1**4 + 448*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1088*K1**3*K3 + 96*K1**2*K2**2*K4 - 2528*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 160*K1**2*K2*K4 + 7616*K1**2*K2 - 1552*K1**2*K3**2 - 96*K1**2*K3*K5 - 224*K1**2*K4**2 - 64*K1**2*K5**2 - 6400*K1**2 - 864*K1*K2**2*K3 - 32*K1*K2**2*K5 + 32*K1*K2*K3**3 - 608*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 7192*K1*K2*K3 - 32*K1*K3**2*K5 + 2792*K1*K3*K4 + 752*K1*K4*K5 + 192*K1*K5*K6 - 64*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 40*K2**2*K4**2 + 1088*K2**2*K4 - 8*K2**2*K6**2 - 5346*K2**2 - 64*K2*K3**2*K4 + 912*K2*K3*K5 + 192*K2*K4*K6 + 8*K2*K5*K7 + 16*K2*K6*K8 - 64*K3**4 - 16*K3**2*K4**2 + 144*K3**2*K6 - 3392*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1458*K4**2 - 580*K5**2 - 174*K6**2 - 12*K7**2 - 12*K8**2 + 5828
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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