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Flat knot 6.172

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,0,3,2,2,4,2,2,1,2,1,1,2,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.172']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 6*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.172', '6.274', '6.286', '6.423', '6.461']
Outer characteristic polynomial of the knot is: t^7+103t^5+104t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.172']
2-strand cable arrow polynomial of the knot is: -976*K1**4 - 256*K1**3*K3 - 128*K1**2*K2**4 + 1280*K1**2*K2**3 - 3696*K1**2*K2**2 - 384*K1**2*K2*K4 + 6200*K1**2*K2 - 208*K1**2*K3**2 - 5584*K1**2 + 928*K1*K2**3*K3 - 1152*K1*K2**2*K3 - 96*K1*K2*K3*K4 + 6016*K1*K2*K3 + 984*K1*K3*K4 + 16*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 288*K2**4*K4 - 1512*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1200*K2**2*K3**2 - 264*K2**2*K4**2 + 1360*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 3660*K2**2 + 752*K2*K3*K5 + 128*K2*K4*K6 + 16*K3**2*K6 - 2316*K3**2 - 640*K4**2 - 172*K5**2 - 36*K6**2 + 4406
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.172']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73350', 'vk6.73396', 'vk6.73512', 'vk6.73573', 'vk6.73727', 'vk6.73844', 'vk6.74260', 'vk6.74886', 'vk6.75335', 'vk6.75518', 'vk6.75844', 'vk6.76433', 'vk6.78234', 'vk6.78319', 'vk6.78478', 'vk6.78643', 'vk6.78836', 'vk6.79304', 'vk6.80057', 'vk6.80101', 'vk6.80206', 'vk6.80269', 'vk6.80399', 'vk6.80765', 'vk6.81960', 'vk6.82687', 'vk6.84751', 'vk6.85047', 'vk6.85145', 'vk6.86513', 'vk6.87345', 'vk6.89425']
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 O1O2O3O4O5U1O6U5U2U4U6U3
R3 orbit {'O1O2O3O4O5U1O6U5U2U4U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U6U2U4U1O6U5
Gauss code of K* O1O2O3O4O5U6U2U5U3U1O6U4
Gauss code of -K* O1O2O3O4O5U2O6U5U3U1U4U6
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 -2 2 1 0 3],[ 4 0 2 4 3 1 3],[ 2 -2 0 3 1 0 3],[-2 -4 -3 0 -1 -1 2],[-1 -3 -1 1 0 0 2],[ 0 -1 0 1 0 0 1],[-3 -3 -3 -2 -2 -1 0]]
Primitive based matrix [[ 0 3 2 1 0 -2 -4],[-3 0 -2 -2 -1 -3 -3],[-2 2 0 -1 -1 -3 -4],[-1 2 1 0 0 -1 -3],[ 0 1 1 0 0 0 -1],[ 2 3 3 1 0 0 -2],[ 4 3 4 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,0,2,4,2,2,1,3,3,1,1,3,4,0,1,3,0,1,2]
Phi over symmetry [-4,-2,0,1,2,3,0,3,2,2,4,2,2,1,2,1,1,2,0,0,-1]
Phi of -K [-4,-2,0,1,2,3,0,3,2,2,4,2,2,1,2,1,1,2,0,0,-1]
Phi of K* [-3,-2,-1,0,2,4,-1,0,2,2,4,0,1,1,2,1,2,2,2,3,0]
Phi of -K* [-4,-2,0,1,2,3,2,1,3,4,3,0,1,3,3,0,1,1,1,2,2]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2-4w^3z+23w^2z+31w
Inner characteristic polynomial t^6+69t^4+28t^2
Outer characteristic polynomial t^7+103t^5+104t^3+8t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 6*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + 2*K2 + 3
2-strand cable arrow polynomial -976*K1**4 - 256*K1**3*K3 - 128*K1**2*K2**4 + 1280*K1**2*K2**3 - 3696*K1**2*K2**2 - 384*K1**2*K2*K4 + 6200*K1**2*K2 - 208*K1**2*K3**2 - 5584*K1**2 + 928*K1*K2**3*K3 - 1152*K1*K2**2*K3 - 96*K1*K2*K3*K4 + 6016*K1*K2*K3 + 984*K1*K3*K4 + 16*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 288*K2**4*K4 - 1512*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1200*K2**2*K3**2 - 264*K2**2*K4**2 + 1360*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 3660*K2**2 + 752*K2*K3*K5 + 128*K2*K4*K6 + 16*K3**2*K6 - 2316*K3**2 - 640*K4**2 - 172*K5**2 - 36*K6**2 + 4406
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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