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Flat knot 6.545

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,0,1,2,0,1,2,2,2,0,0,1,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.545']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**2*K2 + 4*K1**2 - 2*K1*K3 - K2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.133', '6.517', '6.545', '6.1198', '6.1251', '6.1906']
Outer characteristic polynomial of the knot is: t^7+41t^5+83t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.545']
2-strand cable arrow polynomial of the knot is: 1920*K1**2*K2**5 - 3712*K1**2*K2**4 - 640*K1**2*K2**3*K4 + 3296*K1**2*K2**3 - 3872*K1**2*K2**2 - 384*K1**2*K2*K4 + 1872*K1**2*K2 - 1088*K1**2 + 512*K1*K2**5*K3 - 768*K1*K2**4*K3 - 256*K1*K2**4*K5 - 128*K1*K2**3*K3*K4 + 2624*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 1952*K1*K2*K3 + 96*K1*K3*K4 + 96*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1856*K2**6 - 384*K2**4*K3**2 - 192*K2**4*K4**2 + 1408*K2**4*K4 - 1344*K2**4 + 288*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 448*K2**2*K3**2 - 256*K2**2*K4**2 + 1040*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 + 384*K2**2 + 224*K2*K3*K5 + 56*K2*K4*K6 - 272*K3**2 - 146*K4**2 - 64*K5**2 - 8*K6**2 + 760
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.545']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4766', 'vk6.5101', 'vk6.6333', 'vk6.6761', 'vk6.8287', 'vk6.8737', 'vk6.9659', 'vk6.9970', 'vk6.20715', 'vk6.22159', 'vk6.28269', 'vk6.29691', 'vk6.39724', 'vk6.41972', 'vk6.46286', 'vk6.47875', 'vk6.48798', 'vk6.49013', 'vk6.49631', 'vk6.49831', 'vk6.50828', 'vk6.51046', 'vk6.51301', 'vk6.51494', 'vk6.57650', 'vk6.58796', 'vk6.62323', 'vk6.63261', 'vk6.67112', 'vk6.67975', 'vk6.69706', 'vk6.70387']
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 O1O2O3O4O5U3U4U5O6U2U1U6
R3 orbit {'O1O2O3O4O5U3U4U5O6U2U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U5U4O6U1U2U3
Gauss code of K* O1O2O3U4O5O6O4U6U5U1U2U3
Gauss code of -K* O1O2O3U1O4O5O6U4U5U6U3U2
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 -1 -1 -2 0 2 2],[ 1 0 0 -2 0 2 2],[ 1 0 0 -2 0 2 1],[ 2 2 2 0 1 2 0],[ 0 0 0 -1 0 1 0],[-2 -2 -2 -2 -1 0 0],[-2 -2 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 2 0 -1 -1 -2],[-2 0 0 0 -1 -2 0],[-2 0 0 -1 -2 -2 -2],[ 0 0 1 0 0 0 -1],[ 1 1 2 0 0 0 -2],[ 1 2 2 0 0 0 -2],[ 2 0 2 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,1,1,2,0,0,1,2,0,1,2,2,2,0,0,1,0,2,2]
Phi over symmetry [-2,-2,0,1,1,2,0,0,1,2,0,1,2,2,2,0,0,1,0,2,2]
Phi of -K [-2,-1,-1,0,2,2,-1,-1,1,2,4,0,1,1,1,1,1,2,1,2,0]
Phi of K* [-2,-2,0,1,1,2,0,1,1,1,2,2,1,2,4,1,1,1,0,-1,-1]
Phi of -K* [-2,-1,-1,0,2,2,2,2,1,0,2,0,0,1,2,0,2,2,0,1,0]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 3z^2+8z+5
Enhanced Jones-Krushkal polynomial -6w^4z^2+9w^3z^2-10w^3z+18w^2z+5w
Inner characteristic polynomial t^6+27t^4+38t^2
Outer characteristic polynomial t^7+41t^5+83t^3+6t
Flat arrow polynomial -8*K1**4 + 8*K1**2*K2 + 4*K1**2 - 2*K1*K3 - K2
2-strand cable arrow polynomial 1920*K1**2*K2**5 - 3712*K1**2*K2**4 - 640*K1**2*K2**3*K4 + 3296*K1**2*K2**3 - 3872*K1**2*K2**2 - 384*K1**2*K2*K4 + 1872*K1**2*K2 - 1088*K1**2 + 512*K1*K2**5*K3 - 768*K1*K2**4*K3 - 256*K1*K2**4*K5 - 128*K1*K2**3*K3*K4 + 2624*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 1952*K1*K2*K3 + 96*K1*K3*K4 + 96*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1856*K2**6 - 384*K2**4*K3**2 - 192*K2**4*K4**2 + 1408*K2**4*K4 - 1344*K2**4 + 288*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 448*K2**2*K3**2 - 256*K2**2*K4**2 + 1040*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 + 384*K2**2 + 224*K2*K3*K5 + 56*K2*K4*K6 - 272*K3**2 - 146*K4**2 - 64*K5**2 - 8*K6**2 + 760
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{6}, {3, 5}, {4}, {1, 2}]]
If K is slice False
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