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Flat knot 6.848

Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,0,1,2,3,3,0,1,0,0,1,0,0,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.848']
Arrow polynomial of the knot is: 8*K1**3 - 14*K1**2 - 10*K1*K2 - K1 + 7*K2 + 3*K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.848']
Outer characteristic polynomial of the knot is: t^7+47t^5+54t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.848']
2-strand cable arrow polynomial of the knot is: -192*K1**6 - 192*K1**4*K2**2 + 960*K1**4*K2 - 4736*K1**4 + 960*K1**3*K2*K3 + 64*K1**3*K3*K4 - 928*K1**3*K3 - 128*K1**2*K2**4 + 928*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8160*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 12744*K1**2*K2 - 1408*K1**2*K3**2 - 144*K1**2*K4**2 - 7132*K1**2 + 832*K1*K2**3*K3 - 1600*K1*K2**2*K3 - 384*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 10952*K1*K2*K3 + 1936*K1*K3*K4 + 144*K1*K4*K5 + 8*K1*K5*K6 - 192*K2**6 + 320*K2**4*K4 - 1816*K2**4 - 64*K2**3*K6 - 928*K2**2*K3**2 - 136*K2**2*K4**2 + 2080*K2**2*K4 - 5890*K2**2 - 32*K2*K3**2*K4 + 792*K2*K3*K5 + 64*K2*K4*K6 - 32*K3**4 + 48*K3**2*K6 - 3304*K3**2 - 814*K4**2 - 172*K5**2 - 22*K6**2 + 6468
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.848']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16560', 'vk6.16651', 'vk6.18152', 'vk6.18488', 'vk6.22963', 'vk6.23082', 'vk6.24611', 'vk6.25024', 'vk6.34960', 'vk6.35079', 'vk6.36742', 'vk6.37161', 'vk6.42533', 'vk6.42642', 'vk6.44014', 'vk6.44326', 'vk6.54791', 'vk6.54877', 'vk6.55950', 'vk6.56250', 'vk6.59223', 'vk6.59303', 'vk6.60488', 'vk6.60854', 'vk6.64773', 'vk6.64836', 'vk6.65607', 'vk6.65914', 'vk6.68075', 'vk6.68138', 'vk6.68682', 'vk6.68893']
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 O1O2O3O4U1U3O5O6U4U6U2U5
R3 orbit {'O1O2O3O4U1U3O5O6U4U6U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3U6U1O6O5U2U4
Gauss code of K* O1O2O3O4U5U3U6U1O5O6U4U2
Gauss code of -K* O1O2O3O4U3U1O5O6U4U5U2U6
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 -3 0 0 0 2 1],[ 3 0 3 1 2 2 1],[ 0 -3 0 -1 0 2 1],[ 0 -1 1 0 1 1 1],[ 0 -2 0 -1 0 2 1],[-2 -2 -2 -1 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 2 1 0 0 0 -3],[-2 0 0 -1 -2 -2 -2],[-1 0 0 -1 -1 -1 -1],[ 0 1 1 0 1 1 -1],[ 0 2 1 -1 0 0 -2],[ 0 2 1 -1 0 0 -3],[ 3 2 1 1 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,0,3,0,1,2,2,2,1,1,1,1,-1,-1,1,0,2,3]
Phi over symmetry [-3,0,0,0,1,2,0,1,2,3,3,0,1,0,0,1,0,0,0,1,1]
Phi of -K [-3,0,0,0,1,2,0,1,2,3,3,0,1,0,0,1,0,0,0,1,1]
Phi of K* [-2,-1,0,0,0,3,1,0,0,1,3,0,0,0,3,0,-1,0,-1,1,2]
Phi of -K* [-3,0,0,0,1,2,1,2,3,1,2,1,1,1,1,0,1,2,1,2,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+33t^4+15t^2+1
Outer characteristic polynomial t^7+47t^5+54t^3+7t
Flat arrow polynomial 8*K1**3 - 14*K1**2 - 10*K1*K2 - K1 + 7*K2 + 3*K3 + 8
2-strand cable arrow polynomial -192*K1**6 - 192*K1**4*K2**2 + 960*K1**4*K2 - 4736*K1**4 + 960*K1**3*K2*K3 + 64*K1**3*K3*K4 - 928*K1**3*K3 - 128*K1**2*K2**4 + 928*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8160*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 12744*K1**2*K2 - 1408*K1**2*K3**2 - 144*K1**2*K4**2 - 7132*K1**2 + 832*K1*K2**3*K3 - 1600*K1*K2**2*K3 - 384*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 10952*K1*K2*K3 + 1936*K1*K3*K4 + 144*K1*K4*K5 + 8*K1*K5*K6 - 192*K2**6 + 320*K2**4*K4 - 1816*K2**4 - 64*K2**3*K6 - 928*K2**2*K3**2 - 136*K2**2*K4**2 + 2080*K2**2*K4 - 5890*K2**2 - 32*K2*K3**2*K4 + 792*K2*K3*K5 + 64*K2*K4*K6 - 32*K3**4 + 48*K3**2*K6 - 3304*K3**2 - 814*K4**2 - 172*K5**2 - 22*K6**2 + 6468
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}], [{6}, {2, 5}, {4}, {1, 3}]]
If K is slice False
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