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Flat knot 6.863

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,2,1,1,3,0,0,1,1,0,2,2,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.863']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 6*K1**2 - 6*K1*K2 - 2*K1*K3 - 2*K2**2 + 2*K2 + 2*K3 + K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.863']
Outer characteristic polynomial of the knot is: t^7+46t^5+65t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.863']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 672*K1**4*K2 - 2016*K1**4 + 224*K1**3*K2*K3 + 64*K1**3*K3*K4 - 768*K1**3*K3 - 128*K1**2*K2**4 + 256*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 2848*K1**2*K2**2 + 352*K1**2*K2*K3**2 + 128*K1**2*K2*K4**2 - 704*K1**2*K2*K4 + 5744*K1**2*K2 - 1312*K1**2*K3**2 - 64*K1**2*K3*K5 - 336*K1**2*K4**2 - 4308*K1**2 + 384*K1*K2**3*K3 - 736*K1*K2**2*K3 - 128*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 512*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5704*K1*K2*K3 - 192*K1*K2*K4*K5 - 64*K1*K2*K4*K7 - 64*K1*K3**2*K5 - 64*K1*K3*K4*K6 + 2240*K1*K3*K4 + 432*K1*K4*K5 + 64*K1*K5*K6 + 48*K1*K6*K7 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 456*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 576*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 248*K2**2*K4**2 - 32*K2**2*K4*K8 + 944*K2**2*K4 - 8*K2**2*K6**2 - 3272*K2**2 + 592*K2*K3*K5 + 288*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 - 48*K3**2*K4**2 + 136*K3**2*K6 - 2104*K3**2 + 96*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 924*K4**2 - 212*K5**2 - 112*K6**2 - 48*K7**2 - 2*K8**2 + 3716
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.863']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4067', 'vk6.4100', 'vk6.5309', 'vk6.5342', 'vk6.7433', 'vk6.7460', 'vk6.8932', 'vk6.8965', 'vk6.10109', 'vk6.10278', 'vk6.10303', 'vk6.14550', 'vk6.15281', 'vk6.15408', 'vk6.15770', 'vk6.16185', 'vk6.29849', 'vk6.29882', 'vk6.33927', 'vk6.34007', 'vk6.34225', 'vk6.34389', 'vk6.48455', 'vk6.49157', 'vk6.50201', 'vk6.50226', 'vk6.51585', 'vk6.53978', 'vk6.54036', 'vk6.54186', 'vk6.54479', 'vk6.63298']
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 O1O2O3O4U1U4O5O6U5U2U6U3
R3 orbit {'O1O2O3O4U1U4O5O6U5U2U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U3U6O5O6U1U4
Gauss code of K* O1O2O3O4U5U2U4U6O5O6U1U3
Gauss code of -K* O1O2O3O4U2U4O5O6U5U1U3U6
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 -1 2 1 -1 2],[ 3 0 2 3 1 0 1],[ 1 -2 0 2 0 0 2],[-2 -3 -2 0 0 -1 1],[-1 -1 0 0 0 0 0],[ 1 0 0 1 0 0 1],[-2 -1 -2 -1 0 -1 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 1 0 -1 -2 -3],[-2 -1 0 0 -1 -2 -1],[-1 0 0 0 0 0 -1],[ 1 1 1 0 0 0 0],[ 1 2 2 0 0 0 -2],[ 3 3 1 1 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,-1,0,1,2,3,0,1,2,1,0,0,1,0,0,2]
Phi over symmetry [-3,-1,-1,1,2,2,0,2,1,1,3,0,0,1,1,0,2,2,0,0,-1]
Phi of -K [-3,-1,-1,1,2,2,0,2,3,2,4,0,2,1,1,2,2,2,1,1,-1]
Phi of K* [-2,-2,-1,1,1,3,-1,1,1,2,4,1,1,2,2,2,2,3,0,0,2]
Phi of -K* [-3,-1,-1,1,2,2,0,2,1,1,3,0,0,1,1,0,2,2,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2+19w^2z+31w
Inner characteristic polynomial t^6+26t^4+27t^2
Outer characteristic polynomial t^7+46t^5+65t^3+4t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 6*K1**2 - 6*K1*K2 - 2*K1*K3 - 2*K2**2 + 2*K2 + 2*K3 + K4 + 4
2-strand cable arrow polynomial -64*K1**6 + 672*K1**4*K2 - 2016*K1**4 + 224*K1**3*K2*K3 + 64*K1**3*K3*K4 - 768*K1**3*K3 - 128*K1**2*K2**4 + 256*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 2848*K1**2*K2**2 + 352*K1**2*K2*K3**2 + 128*K1**2*K2*K4**2 - 704*K1**2*K2*K4 + 5744*K1**2*K2 - 1312*K1**2*K3**2 - 64*K1**2*K3*K5 - 336*K1**2*K4**2 - 4308*K1**2 + 384*K1*K2**3*K3 - 736*K1*K2**2*K3 - 128*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 512*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5704*K1*K2*K3 - 192*K1*K2*K4*K5 - 64*K1*K2*K4*K7 - 64*K1*K3**2*K5 - 64*K1*K3*K4*K6 + 2240*K1*K3*K4 + 432*K1*K4*K5 + 64*K1*K5*K6 + 48*K1*K6*K7 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 456*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 576*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 248*K2**2*K4**2 - 32*K2**2*K4*K8 + 944*K2**2*K4 - 8*K2**2*K6**2 - 3272*K2**2 + 592*K2*K3*K5 + 288*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 - 48*K3**2*K4**2 + 136*K3**2*K6 - 2104*K3**2 + 96*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 924*K4**2 - 212*K5**2 - 112*K6**2 - 48*K7**2 - 2*K8**2 + 3716
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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