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Flat knot 6.125

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,1,1,3,2,1,1,1,1,1,1,1,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.125', '7.10418']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.121', '6.125', '6.866', '6.894', '6.936', '6.937']
Outer characteristic polynomial of the knot is: t^7+60t^5+81t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.125']
2-strand cable arrow polynomial of the knot is: -512*K1**6 - 1536*K1**4*K2**2 + 2624*K1**4*K2 - 3904*K1**4 + 1664*K1**3*K2*K3 - 672*K1**3*K3 + 384*K1**2*K2**5 - 3456*K1**2*K2**4 + 5728*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 + 448*K1**2*K2**2*K4 - 12752*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 1088*K1**2*K2*K4 + 9840*K1**2*K2 - 768*K1**2*K3**2 - 128*K1**2*K4**2 - 2740*K1**2 + 256*K1*K2**5*K3 - 640*K1*K2**4*K3 - 128*K1*K2**4*K5 + 4640*K1*K2**3*K3 + 608*K1*K2**2*K3*K4 - 2848*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 704*K1*K2**2*K5 + 64*K1*K2*K3**3 - 480*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7944*K1*K2*K3 + 952*K1*K3*K4 + 128*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1600*K2**6 - 128*K2**5*K6 - 704*K2**4*K3**2 - 64*K2**4*K4**2 + 1632*K2**4*K4 - 4336*K2**4 + 416*K2**3*K3*K5 + 64*K2**3*K4*K6 - 192*K2**3*K6 + 64*K2**2*K3**2*K4 - 1744*K2**2*K3**2 - 32*K2**2*K3*K7 - 504*K2**2*K4**2 + 3040*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 730*K2**2 - 32*K2*K3**2*K4 + 696*K2*K3*K5 + 136*K2*K4*K6 - 1160*K3**2 - 410*K4**2 - 60*K5**2 - 6*K6**2 + 2976
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.125']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.488', 'vk6.557', 'vk6.616', 'vk6.962', 'vk6.1058', 'vk6.1118', 'vk6.1650', 'vk6.1761', 'vk6.1833', 'vk6.2143', 'vk6.2238', 'vk6.2302', 'vk6.2577', 'vk6.2843', 'vk6.3054', 'vk6.3174', 'vk6.12048', 'vk6.13039', 'vk6.20487', 'vk6.21000', 'vk6.21841', 'vk6.22422', 'vk6.27884', 'vk6.28452', 'vk6.29393', 'vk6.32690', 'vk6.39322', 'vk6.40221', 'vk6.41500', 'vk6.46720', 'vk6.46873', 'vk6.57349']
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 O1O2O3O4O5O6U3U6U4U5U1U2
R3 orbit {'O1O2O3O4O5O6U3U6U4U5U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U6U2U3U1U4
Gauss code of K* O1O2O3O4O5O6U5U6U1U3U4U2
Gauss code of -K* O1O2O3O4O5O6U5U3U4U6U1U2
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 -3 0 2 1],[ 1 0 1 -3 0 2 1],[-1 -1 0 -3 0 2 1],[ 3 3 3 0 2 3 1],[ 0 0 0 -2 0 1 0],[-2 -2 -2 -3 -1 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -2 -1 -2 -3],[-1 0 0 -1 0 -1 -1],[-1 2 1 0 0 -1 -3],[ 0 1 0 0 0 0 -2],[ 1 2 1 1 0 0 -3],[ 3 3 1 3 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,2,1,2,3,1,0,1,1,0,1,3,0,2,3]
Phi over symmetry [-3,-1,0,1,1,2,-1,1,1,3,2,1,1,1,1,1,1,1,-1,-1,1]
Phi of -K [-3,-1,0,1,1,2,-1,1,1,3,2,1,1,1,1,1,1,1,-1,-1,1]
Phi of K* [-2,-1,-1,0,1,3,-1,1,1,1,2,1,1,1,1,1,1,3,1,1,-1]
Phi of -K* [-3,-1,0,1,1,2,3,2,1,3,3,0,1,1,2,0,0,1,-1,0,2]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 6z^2+25z+27
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2+25w^2z+27w
Inner characteristic polynomial t^6+44t^4+30t^2+1
Outer characteristic polynomial t^7+60t^5+81t^3+10t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -512*K1**6 - 1536*K1**4*K2**2 + 2624*K1**4*K2 - 3904*K1**4 + 1664*K1**3*K2*K3 - 672*K1**3*K3 + 384*K1**2*K2**5 - 3456*K1**2*K2**4 + 5728*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 + 448*K1**2*K2**2*K4 - 12752*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 1088*K1**2*K2*K4 + 9840*K1**2*K2 - 768*K1**2*K3**2 - 128*K1**2*K4**2 - 2740*K1**2 + 256*K1*K2**5*K3 - 640*K1*K2**4*K3 - 128*K1*K2**4*K5 + 4640*K1*K2**3*K3 + 608*K1*K2**2*K3*K4 - 2848*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 704*K1*K2**2*K5 + 64*K1*K2*K3**3 - 480*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7944*K1*K2*K3 + 952*K1*K3*K4 + 128*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1600*K2**6 - 128*K2**5*K6 - 704*K2**4*K3**2 - 64*K2**4*K4**2 + 1632*K2**4*K4 - 4336*K2**4 + 416*K2**3*K3*K5 + 64*K2**3*K4*K6 - 192*K2**3*K6 + 64*K2**2*K3**2*K4 - 1744*K2**2*K3**2 - 32*K2**2*K3*K7 - 504*K2**2*K4**2 + 3040*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 730*K2**2 - 32*K2*K3**2*K4 + 696*K2*K3*K5 + 136*K2*K4*K6 - 1160*K3**2 - 410*K4**2 - 60*K5**2 - 6*K6**2 + 2976
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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