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Flat knot 6.1186

Min(phi) over symmetries of the knot is: [-3,0,1,2,1,2,4,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1186', '7.28051']
Arrow polynomial of the knot is: -6*K1*K2 - 2*K1*K3 + 3*K1 - 2*K2**2 + K2 + 3*K3 + 2*K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.466', '6.871', '6.1186']
Outer characteristic polynomial of the knot is: t^5+29t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1186', '7.28051']
2-strand cable arrow polynomial of the knot is: -528*K1**4 + 288*K1**3*K2*K3 + 32*K1**3*K4*K5 - 608*K1**2*K2**2 + 672*K1**2*K2 - 512*K1**2*K3**2 - 208*K1**2*K4**2 - 128*K1**2*K5**2 - 1112*K1**2 + 1320*K1*K2*K3 + 1184*K1*K3*K4 + 624*K1*K4*K5 + 200*K1*K5*K6 + 8*K1*K6*K7 - 56*K2**2*K4**2 + 560*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 1490*K2**2 + 664*K2*K3*K5 + 104*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 - 16*K3**4 - 64*K3**2*K4**2 + 72*K3**2*K6 - 1288*K3**2 + 64*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 998*K4**2 - 644*K5**2 - 150*K6**2 - 28*K7**2 - 12*K8**2 + 2024
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1186']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3632', 'vk6.3721', 'vk6.3912', 'vk6.4015', 'vk6.7058', 'vk6.7117', 'vk6.7292', 'vk6.7389', 'vk6.11406', 'vk6.12593', 'vk6.12704', 'vk6.19104', 'vk6.19151', 'vk6.19802', 'vk6.25717', 'vk6.25778', 'vk6.26241', 'vk6.26684', 'vk6.31006', 'vk6.31133', 'vk6.32190', 'vk6.37824', 'vk6.37881', 'vk6.44962', 'vk6.48268', 'vk6.48447', 'vk6.50024', 'vk6.50168', 'vk6.52153', 'vk6.63729', 'vk6.66197', 'vk6.66226']
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 O1O2O3O4U1U5U4O6O5U6U3U2
R3 orbit {'O1O2O3O4U1U5U4O6O5U6U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U2U5O6O5U1U6U4
Gauss code of K* O1O2O3U4U2O4O5O6U1U6U5U3
Gauss code of -K* O1O2O3U4U3O5O4O6U5U2U1U6
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 -3 1 1 2 0 -1],[ 3 0 3 2 1 2 0],[-1 -3 0 0 1 -1 -1],[-1 -2 0 0 1 -1 -1],[-2 -1 -1 -1 0 -2 -1],[ 0 -2 1 1 2 0 -1],[ 1 0 1 1 1 1 0]]
Primitive based matrix [[ 0 2 1 0 -3],[-2 0 -1 -2 -1],[-1 1 0 -1 -2],[ 0 2 1 0 -2],[ 3 1 2 2 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,-1,0,3,1,2,1,1,2,2]
Phi over symmetry [-3,0,1,2,1,2,4,0,0,0]
Phi of -K [-3,0,1,2,1,2,4,0,0,0]
Phi of K* [-2,-1,0,3,0,0,4,0,2,1]
Phi of -K* [-3,0,1,2,2,2,1,1,2,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 9z+19
Enhanced Jones-Krushkal polynomial -8w^3z+17w^2z+19w
Inner characteristic polynomial t^4+15t^2+1
Outer characteristic polynomial t^5+29t^3+6t
Flat arrow polynomial -6*K1*K2 - 2*K1*K3 + 3*K1 - 2*K2**2 + K2 + 3*K3 + 2*K4 + 2
2-strand cable arrow polynomial -528*K1**4 + 288*K1**3*K2*K3 + 32*K1**3*K4*K5 - 608*K1**2*K2**2 + 672*K1**2*K2 - 512*K1**2*K3**2 - 208*K1**2*K4**2 - 128*K1**2*K5**2 - 1112*K1**2 + 1320*K1*K2*K3 + 1184*K1*K3*K4 + 624*K1*K4*K5 + 200*K1*K5*K6 + 8*K1*K6*K7 - 56*K2**2*K4**2 + 560*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 1490*K2**2 + 664*K2*K3*K5 + 104*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 - 16*K3**4 - 64*K3**2*K4**2 + 72*K3**2*K6 - 1288*K3**2 + 64*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 998*K4**2 - 644*K5**2 - 150*K6**2 - 28*K7**2 - 12*K8**2 + 2024
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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