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Flat knot 6.22

Min(phi) over symmetries of the knot is: [-5,-2,0,1,3,3,1,4,2,3,5,2,1,2,3,0,2,2,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.22']
Arrow polynomial of the knot is: 4*K1**2*K3 - 2*K1**2 - 6*K1*K2 - 2*K1*K4 + 2*K1 + K2 + 2*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.19', '6.22', '6.41']
Outer characteristic polynomial of the knot is: t^7+132t^5+81t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.22']
2-strand cable arrow polynomial of the knot is: -16*K1**4 + 96*K1**3*K2*K3 - 192*K1**2*K2**2*K3**2 - 976*K1**2*K2**2 + 296*K1**2*K2 - 288*K1**2*K3**2 - 832*K1**2 + 128*K1*K2**3*K3**3 + 1120*K1*K2**3*K3 + 256*K1*K2*K3**3 + 2192*K1*K2*K3 + 200*K1*K3*K4 + 48*K1*K5*K6 - 512*K2**4*K3**2 - 32*K2**4*K6**2 - 584*K2**4 + 320*K2**3*K3*K5 + 64*K2**3*K4*K6 + 32*K2**3*K6*K8 - 128*K2**2*K3**4 - 1264*K2**2*K3**2 - 56*K2**2*K4**2 + 72*K2**2*K4 - 64*K2**2*K5**2 - 64*K2**2*K6**2 - 8*K2**2*K8**2 - 380*K2**2 + 32*K2*K3**3*K5 + 624*K2*K3*K5 + 88*K2*K4*K6 + 8*K2*K5*K7 + 32*K2*K6*K8 - 112*K3**4 + 16*K3**2*K6 - 820*K3**2 + 8*K3*K5*K8 - 86*K4**2 - 140*K5**2 - 60*K6**2 - 12*K8**2 + 992
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.22']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71568', 'vk6.71678', 'vk6.72093', 'vk6.72306', 'vk6.74033', 'vk6.74595', 'vk6.76077', 'vk6.76790', 'vk6.77186', 'vk6.77285', 'vk6.77483', 'vk6.77647', 'vk6.79020', 'vk6.79597', 'vk6.80557', 'vk6.81008', 'vk6.81107', 'vk6.81137', 'vk6.81158', 'vk6.81214', 'vk6.81303', 'vk6.81450', 'vk6.82252', 'vk6.83499', 'vk6.83827', 'vk6.83974', 'vk6.85387', 'vk6.86316', 'vk6.87093', 'vk6.88018', 'vk6.88328', 'vk6.88961']
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 O1O2O3O4O5O6U1U3U5U6U2U4
R3 orbit {'O1O2O3O4O5O6U1U3U5U6U2U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U3U5U1U2U4U6
Gauss code of K* O1O2O3O4O5O6U1U5U2U6U3U4
Gauss code of -K* O1O2O3O4O5O6U3U4U1U5U2U6
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 0 -2 3 1 3],[ 5 0 4 1 5 2 3],[ 0 -4 0 -2 2 0 2],[ 2 -1 2 0 3 1 2],[-3 -5 -2 -3 0 -1 1],[-1 -2 0 -1 1 0 1],[-3 -3 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 3 1 0 -2 -5],[-3 0 1 -1 -2 -3 -5],[-3 -1 0 -1 -2 -2 -3],[-1 1 1 0 0 -1 -2],[ 0 2 2 0 0 -2 -4],[ 2 3 2 1 2 0 -1],[ 5 5 3 2 4 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-1,0,2,5,-1,1,2,3,5,1,2,2,3,0,1,2,2,4,1]
Phi over symmetry [-5,-2,0,1,3,3,1,4,2,3,5,2,1,2,3,0,2,2,1,1,-1]
Phi of -K [-5,-2,0,1,3,3,2,1,4,3,5,0,2,2,3,1,1,1,1,1,-1]
Phi of K* [-3,-3,-1,0,2,5,-1,1,1,3,5,1,1,2,3,1,2,4,0,1,2]
Phi of -K* [-5,-2,0,1,3,3,1,4,2,3,5,2,1,2,3,0,2,2,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^5-2t^3+t^2-t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial -8w^3z+13w^2z+11w
Inner characteristic polynomial t^6+84t^4
Outer characteristic polynomial t^7+132t^5+81t^3
Flat arrow polynomial 4*K1**2*K3 - 2*K1**2 - 6*K1*K2 - 2*K1*K4 + 2*K1 + K2 + 2*K3 + 2
2-strand cable arrow polynomial -16*K1**4 + 96*K1**3*K2*K3 - 192*K1**2*K2**2*K3**2 - 976*K1**2*K2**2 + 296*K1**2*K2 - 288*K1**2*K3**2 - 832*K1**2 + 128*K1*K2**3*K3**3 + 1120*K1*K2**3*K3 + 256*K1*K2*K3**3 + 2192*K1*K2*K3 + 200*K1*K3*K4 + 48*K1*K5*K6 - 512*K2**4*K3**2 - 32*K2**4*K6**2 - 584*K2**4 + 320*K2**3*K3*K5 + 64*K2**3*K4*K6 + 32*K2**3*K6*K8 - 128*K2**2*K3**4 - 1264*K2**2*K3**2 - 56*K2**2*K4**2 + 72*K2**2*K4 - 64*K2**2*K5**2 - 64*K2**2*K6**2 - 8*K2**2*K8**2 - 380*K2**2 + 32*K2*K3**3*K5 + 624*K2*K3*K5 + 88*K2*K4*K6 + 8*K2*K5*K7 + 32*K2*K6*K8 - 112*K3**4 + 16*K3**2*K6 - 820*K3**2 + 8*K3*K5*K8 - 86*K4**2 - 140*K5**2 - 60*K6**2 - 12*K8**2 + 992
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {2, 5}, {3, 4}, {1}]]
If K is slice False
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