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Flat knot 6.251

Min(phi) over symmetries of the knot is: [-4,-2,-1,2,2,3,1,1,2,3,4,0,1,2,2,1,2,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.251']
Arrow polynomial of the knot is: 4*K1**3 - 6*K1*K2 - 2*K1*K3 + K2 + 2*K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.192', '6.251']
Outer characteristic polynomial of the knot is: t^7+119t^5+124t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.251']
2-strand cable arrow polynomial of the knot is: 96*K1**3*K2*K3 - 256*K1**3*K3 + 192*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 1984*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 192*K1**2*K2*K4 + 2808*K1**2*K2 - 1120*K1**2*K3**2 - 64*K1**2*K3*K5 - 3052*K1**2 + 1216*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1536*K1*K2**2*K3 - 192*K1*K2**2*K5 + 128*K1*K2*K3**3 - 704*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5368*K1*K2*K3 - 64*K1*K2*K4*K5 + 1712*K1*K3*K4 + 152*K1*K4*K5 + 48*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**6 + 96*K2**4*K4 - 1328*K2**4 + 32*K2**3*K3*K5 - 32*K2**3*K6 - 1968*K2**2*K3**2 - 120*K2**2*K4**2 + 1584*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 2128*K2**2 - 96*K2*K3**2*K4 + 1240*K2*K3*K5 + 128*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 - 1904*K3**2 - 710*K4**2 - 188*K5**2 - 48*K6**2 - 16*K7**2 - 2*K8**2 + 2590
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.251']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81844', 'vk6.81890', 'vk6.82068', 'vk6.82078', 'vk6.82565', 'vk6.82608', 'vk6.82779', 'vk6.82784', 'vk6.82829', 'vk6.82840', 'vk6.82949', 'vk6.83052', 'vk6.83066', 'vk6.83274', 'vk6.83324', 'vk6.83366', 'vk6.83528', 'vk6.84544', 'vk6.84641', 'vk6.84920', 'vk6.84958', 'vk6.85822', 'vk6.86093', 'vk6.86116', 'vk6.86159', 'vk6.86839', 'vk6.88457', 'vk6.88897', 'vk6.89029', 'vk6.89688', 'vk6.89921', 'vk6.90021']
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 O1O2O3O4O5U1U2O6U3U5U4U6
R3 orbit {'O1O2O3O4O5U1U2O6U3U5U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U2U1U3O6U4U5
Gauss code of K* O1O2O3O4U5U6U1U3U2O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U3U2U4U5U6
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 -1 2 2 3],[ 4 0 1 2 4 3 3],[ 2 -1 0 1 3 2 3],[ 1 -2 -1 0 2 1 3],[-2 -4 -3 -2 0 0 2],[-2 -3 -2 -1 0 0 1],[-3 -3 -3 -3 -2 -1 0]]
Primitive based matrix [[ 0 3 2 2 -1 -2 -4],[-3 0 -1 -2 -3 -3 -3],[-2 1 0 0 -1 -2 -3],[-2 2 0 0 -2 -3 -4],[ 1 3 1 2 0 -1 -2],[ 2 3 2 3 1 0 -1],[ 4 3 3 4 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-2,1,2,4,1,2,3,3,3,0,1,2,3,2,3,4,1,2,1]
Phi over symmetry [-4,-2,-1,2,2,3,1,1,2,3,4,0,1,2,2,1,2,1,0,-1,0]
Phi of -K [-4,-2,-1,2,2,3,1,1,2,3,4,0,1,2,2,1,2,1,0,-1,0]
Phi of K* [-3,-2,-2,1,2,4,-1,0,1,2,4,0,1,1,2,2,2,3,0,1,1]
Phi of -K* [-4,-2,-1,2,2,3,1,2,3,4,3,1,2,3,3,1,2,3,0,1,2]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 4z^2+17z+19
Enhanced Jones-Krushkal polynomial -2w^4z^2+6w^3z^2-6w^3z+23w^2z+19w
Inner characteristic polynomial t^6+81t^4+41t^2
Outer characteristic polynomial t^7+119t^5+124t^3+7t
Flat arrow polynomial 4*K1**3 - 6*K1*K2 - 2*K1*K3 + K2 + 2*K3 + K4 + 1
2-strand cable arrow polynomial 96*K1**3*K2*K3 - 256*K1**3*K3 + 192*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 1984*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 192*K1**2*K2*K4 + 2808*K1**2*K2 - 1120*K1**2*K3**2 - 64*K1**2*K3*K5 - 3052*K1**2 + 1216*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1536*K1*K2**2*K3 - 192*K1*K2**2*K5 + 128*K1*K2*K3**3 - 704*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5368*K1*K2*K3 - 64*K1*K2*K4*K5 + 1712*K1*K3*K4 + 152*K1*K4*K5 + 48*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**6 + 96*K2**4*K4 - 1328*K2**4 + 32*K2**3*K3*K5 - 32*K2**3*K6 - 1968*K2**2*K3**2 - 120*K2**2*K4**2 + 1584*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 2128*K2**2 - 96*K2*K3**2*K4 + 1240*K2*K3*K5 + 128*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 - 1904*K3**2 - 710*K4**2 - 188*K5**2 - 48*K6**2 - 16*K7**2 - 2*K8**2 + 2590
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}]]
If K is slice False
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