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Flat knot 6.186

Min(phi) over symmetries of the knot is: [-4,-3,-1,2,3,3,0,2,3,3,5,1,2,2,3,1,2,3,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.186']
Arrow polynomial of the knot is: -2*K1**2 - 6*K1*K2 + 3*K1 - 2*K2**2 + K2 + 3*K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.186', '6.484']
Outer characteristic polynomial of the knot is: t^7+130t^5+124t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.186']
2-strand cable arrow polynomial of the knot is: -320*K1**4 + 64*K1**3*K2*K3 + 64*K1**3*K3*K4 - 176*K1**2*K2**2 + 312*K1**2*K2 - 1152*K1**2*K3**2 - 176*K1**2*K4**2 - 1588*K1**2 + 32*K1*K2*K3**3 + 2544*K1*K2*K3 + 32*K1*K3**3*K4 + 1936*K1*K3*K4 + 176*K1*K4*K5 + 32*K1*K5*K6 - 8*K2**4 - 96*K2**2*K3**2 - 24*K2**2*K4**2 + 208*K2**2*K4 - 1282*K2**2 + 144*K2*K3*K5 + 72*K2*K4*K6 + 8*K2*K5*K7 - 160*K3**4 - 96*K3**2*K4**2 + 80*K3**2*K6 - 1584*K3**2 + 64*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 838*K4**2 - 124*K5**2 - 54*K6**2 - 16*K7**2 - 2*K8**2 + 1926
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.186']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81566', 'vk6.81581', 'vk6.81646', 'vk6.81663', 'vk6.81743', 'vk6.81744', 'vk6.81852', 'vk6.81854', 'vk6.82234', 'vk6.82245', 'vk6.82397', 'vk6.82398', 'vk6.82512', 'vk6.82513', 'vk6.82568', 'vk6.82570', 'vk6.83182', 'vk6.83186', 'vk6.83599', 'vk6.83600', 'vk6.84132', 'vk6.84145', 'vk6.84352', 'vk6.84353', 'vk6.84560', 'vk6.84562', 'vk6.86472', 'vk6.86488', 'vk6.88740', 'vk6.88741', 'vk6.88911', 'vk6.88913']
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 O1O2O3O4O5U2O6U1U3U5U6U4
R3 orbit {'O1O2O3O4O5U2O6U1U3U5U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U6U1U3U5O6U4
Gauss code of K* O1O2O3O4O5U1U6U2U5U3O6U4
Gauss code of -K* O1O2O3O4O5U2O6U3U1U4U6U5
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 -4 -3 -1 3 2 3],[ 4 0 0 2 5 3 3],[ 3 0 0 1 3 2 2],[ 1 -2 -1 0 3 1 2],[-3 -5 -3 -3 0 -1 1],[-2 -3 -2 -1 1 0 1],[-3 -3 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 3 2 -1 -3 -4],[-3 0 1 -1 -3 -3 -5],[-3 -1 0 -1 -2 -2 -3],[-2 1 1 0 -1 -2 -3],[ 1 3 2 1 0 -1 -2],[ 3 3 2 2 1 0 0],[ 4 5 3 3 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-2,1,3,4,-1,1,3,3,5,1,2,2,3,1,2,3,1,2,0]
Phi over symmetry [-4,-3,-1,2,3,3,0,2,3,3,5,1,2,2,3,1,2,3,1,1,-1]
Phi of -K [-4,-3,-1,2,3,3,1,1,3,2,4,1,3,3,4,2,1,2,0,0,-1]
Phi of K* [-3,-3,-2,1,3,4,-1,0,2,4,4,0,1,3,2,2,3,3,1,1,1]
Phi of -K* [-4,-3,-1,2,3,3,0,2,3,3,5,1,2,2,3,1,2,3,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 11z+23
Enhanced Jones-Krushkal polynomial -4w^3z+15w^2z+23w
Inner characteristic polynomial t^6+82t^4+19t^2
Outer characteristic polynomial t^7+130t^5+124t^3
Flat arrow polynomial -2*K1**2 - 6*K1*K2 + 3*K1 - 2*K2**2 + K2 + 3*K3 + K4 + 3
2-strand cable arrow polynomial -320*K1**4 + 64*K1**3*K2*K3 + 64*K1**3*K3*K4 - 176*K1**2*K2**2 + 312*K1**2*K2 - 1152*K1**2*K3**2 - 176*K1**2*K4**2 - 1588*K1**2 + 32*K1*K2*K3**3 + 2544*K1*K2*K3 + 32*K1*K3**3*K4 + 1936*K1*K3*K4 + 176*K1*K4*K5 + 32*K1*K5*K6 - 8*K2**4 - 96*K2**2*K3**2 - 24*K2**2*K4**2 + 208*K2**2*K4 - 1282*K2**2 + 144*K2*K3*K5 + 72*K2*K4*K6 + 8*K2*K5*K7 - 160*K3**4 - 96*K3**2*K4**2 + 80*K3**2*K6 - 1584*K3**2 + 64*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 838*K4**2 - 124*K5**2 - 54*K6**2 - 16*K7**2 - 2*K8**2 + 1926
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {4, 5}, {1, 2}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice False
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