| Min(phi) over symmetries of the knot is: [-4,0,1,1,2,1,3,4,2,1,1,1,0,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['5.10'] |
| Arrow polynomial of the knot is: -4*K1**2*K2 + 4*K1**2 + 2*K1*K3 - K2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['5.1', '5.10', '7.328', '7.361', '7.789', '7.2224', '7.2289', '7.2579', '7.2621', '7.2984', '7.2999', '7.3003', '7.3307', '7.3342', '7.4820', '7.5074', '7.5342', '7.6103', '7.6227', '7.6233', '7.6332', '7.6341', '7.7236', '7.7266', '7.7269', '7.7303', '7.7837', '7.8068', '7.8129', '7.8142', '7.8194', '7.9334', '7.9364', '7.9377', '7.9465', '7.9474', '7.9486', '7.9662', '7.9899', '7.9930', '7.11901', '7.12302', '7.12348', '7.12492', '7.13054', '7.13065', '7.13105', '7.13161', '7.13190', '7.13192', '7.13246', '7.13401', '7.13405', '7.13815', '7.13863', '7.14615', '7.14740', '7.14744', '7.15091', '7.15105', '7.15129', '7.15133', '7.15189', '7.15955', '7.16089', '7.17594', '7.17655', '7.17680', '7.31413'] |
| Outer characteristic polynomial of the knot is: t^6+57t^4+32t^2+1 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.10', '5.25', '7.21797'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4 - 608*K1**2*K2**2 - 192*K1**2*K2*K4 + 1192*K1**2*K2 - 1080*K1**2 + 224*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 320*K1*K2**2*K3 - 32*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 1264*K1*K2*K3 + 296*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**4*K4**2 + 96*K2**4*K4 - 288*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 288*K2**2*K3**2 - 224*K2**2*K4**2 + 568*K2**2*K4 - 8*K2**2*K6**2 - 852*K2**2 + 104*K2*K3*K5 + 72*K2*K4*K6 - 472*K3**2 - 234*K4**2 - 16*K5**2 - 4*K6**2 + 904 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.10'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.618', 'vk5.622', 'vk5.756', 'vk5.762', 'vk5.1114', 'vk5.1122', 'vk5.1272', 'vk5.1275', 'vk5.1628', 'vk5.1630', 'vk5.1742', 'vk5.1746', 'vk5.1855', 'vk5.1859', 'vk5.1917', 'vk5.1919'] |
| 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 | O1O2O3O4O5U1U4U5U3U2 |
| R3 orbit | {'O1O2O3O4O5U1U4U5U3U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U4U3U1U2U5 |
| Gauss code of K* | O1O2O3O4O5U1U5U4U2U3 |
| Gauss code of -K* | O1O2O3O4O5U3U4U2U1U5 |
| 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 1 1 0 2],[ 4 0 4 3 1 2],[-1 -4 0 0 -1 1],[-1 -3 0 0 -1 1],[ 0 -1 1 1 0 1],[-2 -2 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -4],[-2 0 -1 -1 -1 -2],[-1 1 0 0 -1 -3],[-1 1 0 0 -1 -4],[ 0 1 1 1 0 -1],[ 4 2 3 4 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,4,1,1,1,2,0,1,3,1,4,1] |
| Phi over symmetry | [-4,0,1,1,2,1,3,4,2,1,1,1,0,1,1] |
| Phi of -K | [-4,0,1,1,2,3,1,2,4,0,0,1,0,0,0] |
| Phi of K* | [-2,-1,-1,0,4,0,0,1,4,0,0,1,0,2,3] |
| Phi of -K* | [-4,0,1,1,2,1,3,4,2,1,1,1,0,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^2-2t |
| Normalized Jones-Krushkal polynomial | -3z^2-14z-15 |
| Enhanced Jones-Krushkal polynomial | -3w^3z^2-14w^2z-15w |
| Inner characteristic polynomial | t^5+35t^3+3t |
| Outer characteristic polynomial | t^6+57t^4+32t^2+1 |
| Flat arrow polynomial | -4*K1**2*K2 + 4*K1**2 + 2*K1*K3 - K2 |
| 2-strand cable arrow polynomial | -192*K1**4 - 608*K1**2*K2**2 - 192*K1**2*K2*K4 + 1192*K1**2*K2 - 1080*K1**2 + 224*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 320*K1*K2**2*K3 - 32*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 1264*K1*K2*K3 + 296*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**4*K4**2 + 96*K2**4*K4 - 288*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 288*K2**2*K3**2 - 224*K2**2*K4**2 + 568*K2**2*K4 - 8*K2**2*K6**2 - 852*K2**2 + 104*K2*K3*K5 + 72*K2*K4*K6 - 472*K3**2 - 234*K4**2 - 16*K5**2 - 4*K6**2 + 904 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 5}, {1, 4}, {3}]] |
| If K is slice | False |