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Flat knot 6.904

Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,0,0,3,3,4,0,1,2,2,1,0,1,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.904']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.904']
Outer characteristic polynomial of the knot is: t^7+74t^5+111t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.904']
2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 320*K1**4*K2 - 400*K1**4 + 640*K1**3*K2*K3 - 320*K1**3*K3 + 448*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 256*K1**2*K2**2*K4**2 + 288*K1**2*K2**2*K4 - 3504*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 736*K1**2*K2*K4 + 4200*K1**2*K2 - 320*K1**2*K3**2 - 176*K1**2*K4**2 - 3420*K1**2 + 224*K1*K2**3*K3 + 736*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 + 384*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 + 64*K1*K2*K3**3 + 160*K1*K2*K3*K4**2 - 768*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4552*K1*K2*K3 - 128*K1*K2*K4*K5 - 64*K1*K2*K4*K7 + 1560*K1*K3*K4 + 496*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**4*K4**2 + 96*K2**4*K4 - 400*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 736*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 816*K2**2*K4**2 - 32*K2**2*K4*K8 + 1616*K2**2*K4 - 176*K2**2*K5**2 - 8*K2**2*K6**2 - 2576*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 712*K2*K3*K5 + 304*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 - 16*K3**4 - 32*K3**2*K4**2 + 16*K3**2*K6 - 1596*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1002*K4**2 - 228*K5**2 - 16*K6**2 - 4*K7**2 - 2*K8**2 + 2794
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.904']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4192', 'vk6.4271', 'vk6.5442', 'vk6.5556', 'vk6.7555', 'vk6.7638', 'vk6.9061', 'vk6.9140', 'vk6.18239', 'vk6.18576', 'vk6.24711', 'vk6.25126', 'vk6.36838', 'vk6.37303', 'vk6.44074', 'vk6.44415', 'vk6.48512', 'vk6.48591', 'vk6.49208', 'vk6.49312', 'vk6.50303', 'vk6.50379', 'vk6.51066', 'vk6.51097', 'vk6.56034', 'vk6.56310', 'vk6.60583', 'vk6.60924', 'vk6.65700', 'vk6.65996', 'vk6.68745', 'vk6.68955']
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 O1O2O3O4U2U5O6O5U1U6U3U4
R3 orbit {'O1O2O3O4U2U5O6O5U1U6U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U2U5U4O6O5U6U3
Gauss code of K* O1O2O3O4U1U5U3U4O5O6U2U6
Gauss code of -K* O1O2O3O4U5U3O5O6U1U2U6U4
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 -2 1 3 1 0],[ 3 0 0 3 4 3 0],[ 2 0 0 1 2 2 0],[-1 -3 -1 0 1 0 -1],[-3 -4 -2 -1 0 -2 -1],[-1 -3 -2 0 2 0 0],[ 0 0 0 1 1 0 0]]
Primitive based matrix [[ 0 3 1 1 0 -2 -3],[-3 0 -1 -2 -1 -2 -4],[-1 1 0 0 -1 -1 -3],[-1 2 0 0 0 -2 -3],[ 0 1 1 0 0 0 0],[ 2 2 1 2 0 0 0],[ 3 4 3 3 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,0,2,3,1,2,1,2,4,0,1,1,3,0,2,3,0,0,0]
Phi over symmetry [-3,-2,0,1,1,3,0,0,3,3,4,0,1,2,2,1,0,1,0,1,2]
Phi of -K [-3,-2,0,1,1,3,1,3,1,1,2,2,1,2,3,1,0,2,0,0,1]
Phi of K* [-3,-1,-1,0,2,3,0,1,2,3,2,0,1,1,1,0,2,1,2,3,1]
Phi of -K* [-3,-2,0,1,1,3,0,0,3,3,4,0,1,2,2,1,0,1,0,1,2]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial 6w^3z^2+23w^2z+23w
Inner characteristic polynomial t^6+50t^4+58t^2+1
Outer characteristic polynomial t^7+74t^5+111t^3+5t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3
2-strand cable arrow polynomial -384*K1**4*K2**2 + 320*K1**4*K2 - 400*K1**4 + 640*K1**3*K2*K3 - 320*K1**3*K3 + 448*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 256*K1**2*K2**2*K4**2 + 288*K1**2*K2**2*K4 - 3504*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 736*K1**2*K2*K4 + 4200*K1**2*K2 - 320*K1**2*K3**2 - 176*K1**2*K4**2 - 3420*K1**2 + 224*K1*K2**3*K3 + 736*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 + 384*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 + 64*K1*K2*K3**3 + 160*K1*K2*K3*K4**2 - 768*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4552*K1*K2*K3 - 128*K1*K2*K4*K5 - 64*K1*K2*K4*K7 + 1560*K1*K3*K4 + 496*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**4*K4**2 + 96*K2**4*K4 - 400*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 736*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 816*K2**2*K4**2 - 32*K2**2*K4*K8 + 1616*K2**2*K4 - 176*K2**2*K5**2 - 8*K2**2*K6**2 - 2576*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 712*K2*K3*K5 + 304*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 - 16*K3**4 - 32*K3**2*K4**2 + 16*K3**2*K6 - 1596*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1002*K4**2 - 228*K5**2 - 16*K6**2 - 4*K7**2 - 2*K8**2 + 2794
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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