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Flat knot 6.1920

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,1,2,2,0,0,0,1,1,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1920']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1*K2 - 4*K1*K3 - K1 + K3 + K4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.565', '6.1229', '6.1243', '6.1920']
Outer characteristic polynomial of the knot is: t^7+30t^5+43t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1920']
2-strand cable arrow polynomial of the knot is: -1152*K1**4*K2**2 + 4864*K1**4*K2 - 5856*K1**4 - 384*K1**3*K2**2*K3 + 1536*K1**3*K2*K3 - 1088*K1**3*K3 - 128*K1**2*K2**4 + 2528*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 9488*K1**2*K2**2 + 384*K1**2*K2*K3**2 - 1024*K1**2*K2*K4 + 7280*K1**2*K2 - 1376*K1**2*K3**2 - 128*K1**2*K3*K5 - 224*K1**2*K4**2 - 1648*K1**2 + 576*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1760*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 640*K1*K2**2*K5 - 608*K1*K2*K3*K4 - 320*K1*K2*K3*K6 + 7224*K1*K2*K3 - 96*K1*K2*K4*K5 - 64*K1*K2*K4*K7 - 32*K1*K2*K5*K6 + 1784*K1*K3*K4 + 632*K1*K4*K5 + 144*K1*K5*K6 + 24*K1*K6*K7 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1520*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 128*K2**3*K6 - 688*K2**2*K3**2 - 32*K2**2*K3*K7 - 336*K2**2*K4**2 + 1824*K2**2*K4 - 160*K2**2*K5**2 - 48*K2**2*K6**2 - 2566*K2**2 + 1040*K2*K3*K5 + 432*K2*K4*K6 + 120*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1516*K3**2 - 788*K4**2 - 380*K5**2 - 122*K6**2 - 24*K7**2 - 2*K8**2 + 3044
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1920']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3258', 'vk6.3286', 'vk6.3291', 'vk6.3390', 'vk6.3417', 'vk6.3422', 'vk6.3466', 'vk6.3521', 'vk6.4623', 'vk6.5914', 'vk6.6033', 'vk6.7959', 'vk6.8078', 'vk6.9393', 'vk6.17849', 'vk6.17864', 'vk6.19056', 'vk6.19869', 'vk6.24370', 'vk6.25674', 'vk6.25689', 'vk6.26312', 'vk6.26757', 'vk6.37776', 'vk6.43791', 'vk6.43803', 'vk6.45049', 'vk6.48110', 'vk6.48118', 'vk6.48141', 'vk6.48199', 'vk6.50667']
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 O1O2O3U1U3U4O5O6O4U6U5U2
R3 orbit {'O1O2O3U1U3U4O5O6O4U6U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U5O4O6O5U6U3U2
Gauss code of K* O1O2O3U4U3U5O4O5O6U2U1U6
Gauss code of -K* O1O2O3U2U4U3O4O5O6U1U6U5
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 -2 1 1 2 -1 -1],[ 2 0 2 1 2 0 0],[-1 -2 0 0 1 -1 -1],[-1 -1 0 0 -1 0 0],[-2 -2 -1 1 0 -2 -1],[ 1 0 1 0 2 0 0],[ 1 0 1 0 1 0 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 1 -1 -1 -2 -2],[-1 -1 0 0 0 0 -1],[-1 1 0 0 -1 -1 -2],[ 1 1 0 1 0 0 0],[ 1 2 0 1 0 0 0],[ 2 2 1 2 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,-1,1,1,2,2,0,0,0,1,1,1,2,0,0,0]
Phi over symmetry [-2,-1,-1,1,1,2,-1,1,1,2,2,0,0,0,1,1,1,2,0,0,0]
Phi of -K [-2,-1,-1,1,1,2,1,1,1,2,2,0,1,2,1,1,2,2,0,0,2]
Phi of K* [-2,-1,-1,1,1,2,0,2,1,2,2,0,1,1,1,2,2,2,0,1,1]
Phi of -K* [-2,-1,-1,1,1,2,0,0,1,2,2,0,0,1,1,0,1,2,0,-1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+18t^4+23t^2+1
Outer characteristic polynomial t^7+30t^5+43t^3+4t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 4*K1*K2 - 4*K1*K3 - K1 + K3 + K4
2-strand cable arrow polynomial -1152*K1**4*K2**2 + 4864*K1**4*K2 - 5856*K1**4 - 384*K1**3*K2**2*K3 + 1536*K1**3*K2*K3 - 1088*K1**3*K3 - 128*K1**2*K2**4 + 2528*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 9488*K1**2*K2**2 + 384*K1**2*K2*K3**2 - 1024*K1**2*K2*K4 + 7280*K1**2*K2 - 1376*K1**2*K3**2 - 128*K1**2*K3*K5 - 224*K1**2*K4**2 - 1648*K1**2 + 576*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1760*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 640*K1*K2**2*K5 - 608*K1*K2*K3*K4 - 320*K1*K2*K3*K6 + 7224*K1*K2*K3 - 96*K1*K2*K4*K5 - 64*K1*K2*K4*K7 - 32*K1*K2*K5*K6 + 1784*K1*K3*K4 + 632*K1*K4*K5 + 144*K1*K5*K6 + 24*K1*K6*K7 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1520*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 128*K2**3*K6 - 688*K2**2*K3**2 - 32*K2**2*K3*K7 - 336*K2**2*K4**2 + 1824*K2**2*K4 - 160*K2**2*K5**2 - 48*K2**2*K6**2 - 2566*K2**2 + 1040*K2*K3*K5 + 432*K2*K4*K6 + 120*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1516*K3**2 - 788*K4**2 - 380*K5**2 - 122*K6**2 - 24*K7**2 - 2*K8**2 + 3044
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice False
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