Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,-1,-1,2,2,3,0,2,1,2,1,1,1,1,0,-1] |
Flat knots (up to 7 crossings) with same phi are :['6.389', '7.33541'] |
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.315', '6.337', '6.389', '6.418', '6.599', '6.675', '6.686', '6.688', '6.746', '6.747', '6.809', '6.1034', '6.1128', '6.1133', '6.1334', '6.1363', '6.1489', '6.1539', '6.1564', '6.1821', '6.1863'] |
Outer characteristic polynomial of the knot is: t^7+51t^5+34t^3+5t |
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.389'] |
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 512*K1**4*K2 - 928*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 + 448*K1**2*K2**3 - 3792*K1**2*K2**2 - 576*K1**2*K2*K4 + 5016*K1**2*K2 - 160*K1**2*K3**2 - 144*K1**2*K4**2 - 3776*K1**2 + 224*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 192*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 3648*K1*K2*K3 + 832*K1*K3*K4 + 136*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 592*K2**4 - 272*K2**2*K3**2 - 176*K2**2*K4**2 + 912*K2**2*K4 - 2518*K2**2 + 104*K2*K3*K5 + 48*K2*K4*K6 - 1056*K3**2 - 564*K4**2 - 56*K5**2 - 2*K6**2 + 2858 |
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.389'] |
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11284', 'vk6.11362', 'vk6.12545', 'vk6.12656', 'vk6.18367', 'vk6.18706', 'vk6.24813', 'vk6.25272', 'vk6.30966', 'vk6.31091', 'vk6.32144', 'vk6.32263', 'vk6.36995', 'vk6.37446', 'vk6.44174', 'vk6.44494', 'vk6.52036', 'vk6.52123', 'vk6.52879', 'vk6.52944', 'vk6.56139', 'vk6.56366', 'vk6.60658', 'vk6.61005', 'vk6.63657', 'vk6.63702', 'vk6.64085', 'vk6.64130', 'vk6.65793', 'vk6.66050', 'vk6.68792', 'vk6.69001'] |
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 | O1O2O3O4O5U4U2O6U5U1U6U3 |
R3 orbit | {'O1O2O3O4O5U4U2O6U5U1U6U3'} |
R3 orbit length | 1 |
Gauss code of -K | O1O2O3O4O5U3U6U5U1O6U4U2 |
Gauss code of K* | O1O2O3O4U2U5U4U6U1O6O5U3 |
Gauss code of -K* | O1O2O3O4U2O5O6U4U6U1U5U3 |
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 2 -1 1 2],[ 2 0 -1 3 -1 2 2],[ 2 1 0 2 0 2 1],[-2 -3 -2 0 -1 0 1],[ 1 1 0 1 0 1 1],[-1 -2 -2 0 -1 0 1],[-2 -2 -1 -1 -1 -1 0]] |
Primitive based matrix | [[ 0 2 2 1 -1 -2 -2],[-2 0 1 0 -1 -2 -3],[-2 -1 0 -1 -1 -1 -2],[-1 0 1 0 -1 -2 -2],[ 1 1 1 1 0 0 1],[ 2 2 1 2 0 0 1],[ 2 3 2 2 -1 -1 0]] |
If based matrix primitive | True |
Phi of primitive based matrix | [-2,-2,-1,1,2,2,-1,0,1,2,3,1,1,1,2,1,2,2,0,-1,-1] |
Phi over symmetry | [-2,-2,-1,1,2,2,-1,-1,2,2,3,0,2,1,2,1,1,1,1,0,-1] |
Phi of -K | [-2,-2,-1,1,2,2,-1,1,1,2,3,2,1,1,2,1,2,2,1,0,-1] |
Phi of K* | [-2,-2,-1,1,2,2,-1,0,2,2,3,1,2,1,2,1,1,1,2,1,-1] |
Phi of -K* | [-2,-2,-1,1,2,2,-1,-1,2,2,3,0,2,1,2,1,1,1,1,0,-1] |
Symmetry type of based matrix | c |
u-polynomial | 0 |
Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
Enhanced Jones-Krushkal polynomial | 4w^3z^2+21w^2z+27w |
Inner characteristic polynomial | t^6+33t^4+16t^2+1 |
Outer characteristic polynomial | t^7+51t^5+34t^3+5t |
Flat arrow polynomial | 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
2-strand cable arrow polynomial | -192*K1**4*K2**2 + 512*K1**4*K2 - 928*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 + 448*K1**2*K2**3 - 3792*K1**2*K2**2 - 576*K1**2*K2*K4 + 5016*K1**2*K2 - 160*K1**2*K3**2 - 144*K1**2*K4**2 - 3776*K1**2 + 224*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 192*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 3648*K1*K2*K3 + 832*K1*K3*K4 + 136*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 592*K2**4 - 272*K2**2*K3**2 - 176*K2**2*K4**2 + 912*K2**2*K4 - 2518*K2**2 + 104*K2*K3*K5 + 48*K2*K4*K6 - 1056*K3**2 - 564*K4**2 - 56*K5**2 - 2*K6**2 + 2858 |
Genus of based matrix | 0 |
Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}]] |
If K is slice | True |