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Flat knot 6.17

Min(phi) over symmetries of the knot is: [-5,-2,0,1,2,4,1,2,5,3,4,1,3,2,3,1,1,2,1,3,1]
Flat knots (up to 7 crossings) with same phi are :['6.17']
Arrow polynomial of the knot is: 16*K1**5 - 8*K1**4 - 8*K1**3*K2 - 12*K1**3 + 4*K1**2*K2 + 4*K1**2 + 2*K1*K2 + 2*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.17']
Outer characteristic polynomial of the knot is: t^7+145t^5+70t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.17']
2-strand cable arrow polynomial of the knot is: -96*K1**4 + 1024*K1**2*K2**7 - 3328*K1**2*K2**6 + 1152*K1**2*K2**5 + 192*K1**2*K2**4 + 32*K1**2*K2**3 - 672*K1**2*K2**2 + 752*K1**2*K2 - 32*K1**2*K3**2 - 632*K1**2 + 512*K1*K2**7*K3 + 1152*K1*K2**5*K3 - 96*K1*K2**3*K3 + 536*K1*K2*K3 + 96*K1*K3*K4 + 16*K1*K4*K5 + 8*K1*K5*K6 - 512*K2**10 + 512*K2**8*K4 - 896*K2**8 - 512*K2**6*K3**2 - 128*K2**6*K4**2 + 928*K2**6 + 128*K2**5*K3*K5 + 64*K2**4*K3**2 + 32*K2**4*K4**2 + 64*K2**4*K4 - 272*K2**4 - 32*K2**3*K3*K5 - 96*K2**2*K3**2 - 24*K2**2*K4**2 + 160*K2**2*K4 - 264*K2**2 + 48*K2*K3*K5 + 8*K2*K4*K6 - 192*K3**2 - 100*K4**2 - 24*K5**2 - 8*K6**2 + 498
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.17']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19908', 'vk6.19910', 'vk6.21123', 'vk6.21126', 'vk6.26811', 'vk6.26819', 'vk6.28601', 'vk6.28604', 'vk6.38245', 'vk6.38253', 'vk6.40359', 'vk6.40370', 'vk6.45108', 'vk6.45116', 'vk6.46977', 'vk6.46980', 'vk6.56678', 'vk6.56682', 'vk6.57749', 'vk6.57756', 'vk6.61049', 'vk6.61065', 'vk6.62333', 'vk6.62339', 'vk6.66375', 'vk6.66377', 'vk6.67127', 'vk6.67134', 'vk6.69030', 'vk6.69032', 'vk6.69825', 'vk6.69827']
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 O1O2O3O4O5O6U1U3U4U5U6U2
R3 orbit {'O1O2O3O4O5O6U1U3U4U5U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U1U2U3U4U6
Gauss code of K* O1O2O3O4O5O6U1U6U2U3U4U5
Gauss code of -K* O1O2O3O4O5O6U2U3U4U5U1U6
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -5 1 -2 0 2 4],[ 5 0 5 1 2 3 4],[-1 -5 0 -3 -1 1 3],[ 2 -1 3 0 1 2 3],[ 0 -2 1 -1 0 1 2],[-2 -3 -1 -2 -1 0 1],[-4 -4 -3 -3 -2 -1 0]]
Primitive based matrix [[ 0 4 2 1 0 -2 -5],[-4 0 -1 -3 -2 -3 -4],[-2 1 0 -1 -1 -2 -3],[-1 3 1 0 -1 -3 -5],[ 0 2 1 1 0 -1 -2],[ 2 3 2 3 1 0 -1],[ 5 4 3 5 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-1,0,2,5,1,3,2,3,4,1,1,2,3,1,3,5,1,2,1]
Phi over symmetry [-5,-2,0,1,2,4,1,2,5,3,4,1,3,2,3,1,1,2,1,3,1]
Phi of -K [-5,-2,0,1,2,4,2,3,1,4,5,1,0,2,3,0,1,2,0,0,1]
Phi of K* [-4,-2,-1,0,2,5,1,0,2,3,5,0,1,2,4,0,0,1,1,3,2]
Phi of -K* [-5,-2,0,1,2,4,1,2,5,3,4,1,3,2,3,1,1,2,1,3,1]
Symmetry type of based matrix c
u-polynomial t^5-t^4-t
Normalized Jones-Krushkal polynomial z+3
Enhanced Jones-Krushkal polynomial -8w^5z+16w^4z-6w^3z-w^2z+3w
Inner characteristic polynomial t^6+95t^4
Outer characteristic polynomial t^7+145t^5+70t^3
Flat arrow polynomial 16*K1**5 - 8*K1**4 - 8*K1**3*K2 - 12*K1**3 + 4*K1**2*K2 + 4*K1**2 + 2*K1*K2 + 2*K1 + 1
2-strand cable arrow polynomial -96*K1**4 + 1024*K1**2*K2**7 - 3328*K1**2*K2**6 + 1152*K1**2*K2**5 + 192*K1**2*K2**4 + 32*K1**2*K2**3 - 672*K1**2*K2**2 + 752*K1**2*K2 - 32*K1**2*K3**2 - 632*K1**2 + 512*K1*K2**7*K3 + 1152*K1*K2**5*K3 - 96*K1*K2**3*K3 + 536*K1*K2*K3 + 96*K1*K3*K4 + 16*K1*K4*K5 + 8*K1*K5*K6 - 512*K2**10 + 512*K2**8*K4 - 896*K2**8 - 512*K2**6*K3**2 - 128*K2**6*K4**2 + 928*K2**6 + 128*K2**5*K3*K5 + 64*K2**4*K3**2 + 32*K2**4*K4**2 + 64*K2**4*K4 - 272*K2**4 - 32*K2**3*K3*K5 - 96*K2**2*K3**2 - 24*K2**2*K4**2 + 160*K2**2*K4 - 264*K2**2 + 48*K2*K3*K5 + 8*K2*K4*K6 - 192*K3**2 - 100*K4**2 - 24*K5**2 - 8*K6**2 + 498
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {5}, {2, 3}, {1}]]
If K is slice False
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