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Flat knot 6.925

Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,0,-1,2,1,2,-1,2,2,3,1,1,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.925']
Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 + 2*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.540', '6.925', '6.1021', '6.1117', '6.1120', '6.1135', '6.1227', '6.1230', '6.1260', '6.1682', '6.1685', '6.1922']
Outer characteristic polynomial of the knot is: t^7+55t^5+59t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.925']
2-strand cable arrow polynomial of the knot is: 128*K1**4*K2 - 2096*K1**4 + 96*K1**3*K3*K4 - 640*K1**3*K3 - 256*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 4000*K1**2*K2 - 1104*K1**2*K3**2 - 224*K1**2*K3*K5 - 656*K1**2*K4**2 - 3928*K1**2 + 64*K1*K2**3*K3 - 320*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 3912*K1*K2*K3 + 3176*K1*K3*K4 + 768*K1*K4*K5 - 32*K2**4 - 128*K2**2*K3**2 - 16*K2**2*K4**2 + 464*K2**2*K4 - 2764*K2**2 + 232*K2*K3*K5 + 32*K2*K4*K6 - 2420*K3**2 - 1364*K4**2 - 252*K5**2 - 12*K6**2 + 3690
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.925']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.14106', 'vk6.14107', 'vk6.14317', 'vk6.14318', 'vk6.15547', 'vk6.15548', 'vk6.16029', 'vk6.16030', 'vk6.16440', 'vk6.16451', 'vk6.16455', 'vk6.22850', 'vk6.22854', 'vk6.34060', 'vk6.34121', 'vk6.34459', 'vk6.34497', 'vk6.34799', 'vk6.34823', 'vk6.34827', 'vk6.42421', 'vk6.42425', 'vk6.54071', 'vk6.54072', 'vk6.54291', 'vk6.54292', 'vk6.54678', 'vk6.54702', 'vk6.54706', 'vk6.64529', 'vk6.64530', 'vk6.64741']
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 O1O2O3O4U3U5O6O5U2U1U4U6
R3 orbit {'O1O2O3O4U3U5O6O5U2U1U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1U4U3O6O5U6U2
Gauss code of K* O1O2O3O4U2U1U5U3O5O6U4U6
Gauss code of -K* O1O2O3O4U5U1O5O6U2U6U4U3
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 -2 -1 2 1 2],[ 2 0 0 0 3 2 2],[ 2 0 0 0 2 2 1],[ 1 0 0 0 1 1 1],[-2 -3 -2 -1 0 -2 0],[-1 -2 -2 -1 2 0 2],[-2 -2 -1 -1 0 -2 0]]
Primitive based matrix [[ 0 2 2 1 -1 -2 -2],[-2 0 0 -2 -1 -1 -2],[-2 0 0 -2 -1 -2 -3],[-1 2 2 0 -1 -2 -2],[ 1 1 1 1 0 0 0],[ 2 1 2 2 0 0 0],[ 2 2 3 2 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,2,2,0,2,1,1,2,2,1,2,3,1,2,2,0,0,0]
Phi over symmetry [-2,-2,-1,1,2,2,0,-1,2,1,2,-1,2,2,3,1,1,1,1,1,0]
Phi of -K [-2,-2,-1,1,2,2,0,1,1,1,2,1,1,2,3,1,2,2,-1,-1,0]
Phi of K* [-2,-2,-1,1,2,2,0,-1,2,1,2,-1,2,2,3,1,1,1,1,1,0]
Phi of -K* [-2,-2,-1,1,2,2,0,0,2,1,2,0,2,2,3,1,1,1,2,2,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+37t^4+21t^2+1
Outer characteristic polynomial t^7+55t^5+59t^3+5t
Flat arrow polynomial -4*K1*K2 + 2*K1 + 2*K3 + 1
2-strand cable arrow polynomial 128*K1**4*K2 - 2096*K1**4 + 96*K1**3*K3*K4 - 640*K1**3*K3 - 256*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 4000*K1**2*K2 - 1104*K1**2*K3**2 - 224*K1**2*K3*K5 - 656*K1**2*K4**2 - 3928*K1**2 + 64*K1*K2**3*K3 - 320*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 3912*K1*K2*K3 + 3176*K1*K3*K4 + 768*K1*K4*K5 - 32*K2**4 - 128*K2**2*K3**2 - 16*K2**2*K4**2 + 464*K2**2*K4 - 2764*K2**2 + 232*K2*K3*K5 + 32*K2*K4*K6 - 2420*K3**2 - 1364*K4**2 - 252*K5**2 - 12*K6**2 + 3690
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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