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Flat knot 6.6

Min(phi) over symmetries of the knot is: [-5,-3,0,1,3,4,1,2,4,3,5,1,3,2,4,1,1,2,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.6']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 6*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1*K4 - 2*K1 + 2*K2 + K3 + K5 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.6']
Outer characteristic polynomial of the knot is: t^7+156t^5+125t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.6']
2-strand cable arrow polynomial of the knot is: -336*K1**4 + 160*K1**3*K2*K3 - 128*K1**2*K2**4 + 256*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 1440*K1**2*K2**2 + 1536*K1**2*K2 - 432*K1**2*K3**2 - 32*K1**2*K4**2 - 1456*K1**2 + 128*K1*K2**3*K3**3 + 832*K1*K2**3*K3 + 352*K1*K2*K3**3 + 1952*K1*K2*K3 + 336*K1*K3*K4 + 112*K1*K4*K5 + 16*K1*K6*K7 + 8*K1*K7*K8 - 2*K10**2 + 8*K10*K2*K8 - 32*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 792*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 + 32*K2**3*K5*K7 - 128*K2**2*K3**4 - 928*K2**2*K3**2 - 200*K2**2*K4**2 + 472*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 824*K2**2 + 32*K2*K3**3*K5 + 416*K2*K3*K5 + 104*K2*K4*K6 + 88*K2*K5*K7 + 8*K2*K6*K8 + 8*K2*K7*K9 - 160*K3**4 + 32*K3**2*K6 - 584*K3**2 + 8*K3*K5*K8 - 308*K4**2 - 128*K5**2 - 38*K6**2 - 40*K7**2 - 12*K8**2 + 1438
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.6']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81558', 'vk6.81632', 'vk6.81636', 'vk6.81831', 'vk6.81839', 'vk6.82049', 'vk6.82216', 'vk6.82220', 'vk6.82326', 'vk6.82330', 'vk6.82537', 'vk6.82545', 'vk6.82986', 'vk6.83116', 'vk6.83118', 'vk6.83547', 'vk6.83549', 'vk6.83920', 'vk6.84072', 'vk6.84076', 'vk6.84527', 'vk6.84885', 'vk6.84893', 'vk6.85892', 'vk6.85894', 'vk6.86397', 'vk6.86401', 'vk6.86439', 'vk6.86441', 'vk6.88829', 'vk6.89762', 'vk6.89880']
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 O1O2O3O4O5O6U1U2U4U6U3U5
R3 orbit {'O1O2O3O4O5O6U1U2U4U6U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U2U4U1U3U5U6
Gauss code of K* O1O2O3O4O5O6U1U2U5U3U6U4
Gauss code of -K* O1O2O3O4O5O6U3U1U4U2U5U6
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -5 -3 1 0 4 3],[ 5 0 1 4 2 5 3],[ 3 -1 0 3 1 4 2],[-1 -4 -3 0 -1 2 1],[ 0 -2 -1 1 0 2 1],[-4 -5 -4 -2 -2 0 0],[-3 -3 -2 -1 -1 0 0]]
Primitive based matrix [[ 0 4 3 1 0 -3 -5],[-4 0 0 -2 -2 -4 -5],[-3 0 0 -1 -1 -2 -3],[-1 2 1 0 -1 -3 -4],[ 0 2 1 1 0 -1 -2],[ 3 4 2 3 1 0 -1],[ 5 5 3 4 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-3,-1,0,3,5,0,2,2,4,5,1,1,2,3,1,3,4,1,2,1]
Phi over symmetry [-5,-3,0,1,3,4,1,2,4,3,5,1,3,2,4,1,1,2,1,2,0]
Phi of -K [-5,-3,0,1,3,4,1,3,2,5,4,2,1,4,3,0,2,2,1,1,1]
Phi of K* [-4,-3,-1,0,3,5,1,1,2,3,4,1,2,4,5,0,1,2,2,3,1]
Phi of -K* [-5,-3,0,1,3,4,1,2,4,3,5,1,3,2,4,1,1,2,1,2,0]
Symmetry type of based matrix c
u-polynomial t^5-t^4-t
Normalized Jones-Krushkal polynomial 9z+19
Enhanced Jones-Krushkal polynomial -4w^3z+13w^2z+19w
Inner characteristic polynomial t^6+96t^4+15t^2
Outer characteristic polynomial t^7+156t^5+125t^3
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 6*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1*K4 - 2*K1 + 2*K2 + K3 + K5 + 3
2-strand cable arrow polynomial -336*K1**4 + 160*K1**3*K2*K3 - 128*K1**2*K2**4 + 256*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 1440*K1**2*K2**2 + 1536*K1**2*K2 - 432*K1**2*K3**2 - 32*K1**2*K4**2 - 1456*K1**2 + 128*K1*K2**3*K3**3 + 832*K1*K2**3*K3 + 352*K1*K2*K3**3 + 1952*K1*K2*K3 + 336*K1*K3*K4 + 112*K1*K4*K5 + 16*K1*K6*K7 + 8*K1*K7*K8 - 2*K10**2 + 8*K10*K2*K8 - 32*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 792*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 + 32*K2**3*K5*K7 - 128*K2**2*K3**4 - 928*K2**2*K3**2 - 200*K2**2*K4**2 + 472*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 824*K2**2 + 32*K2*K3**3*K5 + 416*K2*K3*K5 + 104*K2*K4*K6 + 88*K2*K5*K7 + 8*K2*K6*K8 + 8*K2*K7*K9 - 160*K3**4 + 32*K3**2*K6 - 584*K3**2 + 8*K3*K5*K8 - 308*K4**2 - 128*K5**2 - 38*K6**2 - 40*K7**2 - 12*K8**2 + 1438
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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