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Flat knot 6.156

Min(phi) over symmetries of the knot is: [-4,-2,1,1,2,2,1,1,3,3,4,0,2,1,2,0,0,-1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.156']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 3*K2 + K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.156', '6.476', '6.483']
Outer characteristic polynomial of the knot is: t^7+95t^5+103t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.156']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 672*K1**4*K2 - 2208*K1**4 + 480*K1**3*K2*K3 + 32*K1**3*K3*K4 - 448*K1**3*K3 - 192*K1**2*K2**4 + 416*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 4320*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 608*K1**2*K2*K4 + 7640*K1**2*K2 - 848*K1**2*K3**2 - 32*K1**2*K3*K5 - 144*K1**2*K4**2 - 5740*K1**2 + 640*K1*K2**3*K3 - 1248*K1*K2**2*K3 - 64*K1*K2**2*K5 + 64*K1*K2*K3**3 - 416*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6952*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1824*K1*K3*K4 + 296*K1*K4*K5 + 32*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**6 + 64*K2**4*K4 - 688*K2**4 - 32*K2**3*K6 - 672*K2**2*K3**2 - 48*K2**2*K4**2 + 1280*K2**2*K4 - 8*K2**2*K6**2 - 4566*K2**2 - 64*K2*K3**2*K4 + 496*K2*K3*K5 + 224*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 72*K3**2*K6 - 2548*K3**2 + 24*K3*K4*K7 - 958*K4**2 - 176*K5**2 - 114*K6**2 - 16*K7**2 - 2*K8**2 + 4870
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.156']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16955', 'vk6.17197', 'vk6.20544', 'vk6.21943', 'vk6.23352', 'vk6.23646', 'vk6.27998', 'vk6.29463', 'vk6.35397', 'vk6.35816', 'vk6.39402', 'vk6.41593', 'vk6.42870', 'vk6.43147', 'vk6.45978', 'vk6.47652', 'vk6.55106', 'vk6.55363', 'vk6.57408', 'vk6.58581', 'vk6.59505', 'vk6.59800', 'vk6.62075', 'vk6.63055', 'vk6.64949', 'vk6.65156', 'vk6.66948', 'vk6.67807', 'vk6.68239', 'vk6.68381', 'vk6.69559', 'vk6.70254']
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 O1O2O3O4O5U1O6U3U5U6U4U2
R3 orbit {'O1O2O3O4O5U1O6U3U5U6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U2U6U1U3O6U5
Gauss code of K* O1O2O3O4O5U6U5U1U4U2O6U3
Gauss code of -K* O1O2O3O4O5U3O6U4U2U5U1U6
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 -4 1 -2 2 1 2],[ 4 0 4 1 3 2 2],[-1 -4 0 -3 1 0 2],[ 2 -1 3 0 3 1 2],[-2 -3 -1 -3 0 -1 1],[-1 -2 0 -1 1 0 1],[-2 -2 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 2 2 1 1 -2 -4],[-2 0 1 -1 -1 -3 -3],[-2 -1 0 -1 -2 -2 -2],[-1 1 1 0 0 -1 -2],[-1 1 2 0 0 -3 -4],[ 2 3 2 1 3 0 -1],[ 4 3 2 2 4 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,-1,2,4,-1,1,1,3,3,1,2,2,2,0,1,2,3,4,1]
Phi over symmetry [-4,-2,1,1,2,2,1,1,3,3,4,0,2,1,2,0,0,-1,0,0,-1]
Phi of -K [-4,-2,1,1,2,2,1,1,3,3,4,0,2,1,2,0,0,-1,0,0,-1]
Phi of K* [-2,-2,-1,-1,2,4,-1,-1,0,2,4,0,0,1,3,0,0,1,2,3,1]
Phi of -K* [-4,-2,1,1,2,2,1,2,4,2,3,1,3,2,3,0,1,1,2,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 3z^2+22z+33
Enhanced Jones-Krushkal polynomial 3w^3z^2+22w^2z+33w
Inner characteristic polynomial t^6+65t^4+26t^2
Outer characteristic polynomial t^7+95t^5+103t^3+5t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 3*K2 + K3 + K4 + 3
2-strand cable arrow polynomial -64*K1**6 + 672*K1**4*K2 - 2208*K1**4 + 480*K1**3*K2*K3 + 32*K1**3*K3*K4 - 448*K1**3*K3 - 192*K1**2*K2**4 + 416*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 4320*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 608*K1**2*K2*K4 + 7640*K1**2*K2 - 848*K1**2*K3**2 - 32*K1**2*K3*K5 - 144*K1**2*K4**2 - 5740*K1**2 + 640*K1*K2**3*K3 - 1248*K1*K2**2*K3 - 64*K1*K2**2*K5 + 64*K1*K2*K3**3 - 416*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6952*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1824*K1*K3*K4 + 296*K1*K4*K5 + 32*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**6 + 64*K2**4*K4 - 688*K2**4 - 32*K2**3*K6 - 672*K2**2*K3**2 - 48*K2**2*K4**2 + 1280*K2**2*K4 - 8*K2**2*K6**2 - 4566*K2**2 - 64*K2*K3**2*K4 + 496*K2*K3*K5 + 224*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 72*K3**2*K6 - 2548*K3**2 + 24*K3*K4*K7 - 958*K4**2 - 176*K5**2 - 114*K6**2 - 16*K7**2 - 2*K8**2 + 4870
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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