| Min(phi) over symmetries of the knot is: [-2,-1,0,1,2,0,1,0,2,0,1,0,0,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['5.90', '6.1310'] |
| Arrow polynomial of the knot is: -8*K1**3 + 4*K1*K2 + 4*K1 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['5.21', '5.90', '7.937', '7.5242', '7.10240', '7.10424', '7.15087', '7.16065', '7.16077', '7.16901', '7.19762', '7.19809', '7.20215', '7.20449', '7.23707', '7.24161', '7.24266', '7.24299', '7.26838', '7.29479', '7.29748', '7.29900', '7.29919', '7.30141', '7.30206', '7.30880', '7.31311', '7.31314', '7.32243', '7.32509', '7.32630', '7.32734', '7.34203', '7.34609', '7.37116', '7.37160', '7.37163', '7.37185', '7.37341', '7.37355', '7.39826', '7.40185', '7.40486', '7.41105', '7.41195', '7.41237', '7.41713', '7.41753', '7.41777', '7.43960'] |
| Outer characteristic polynomial of the knot is: t^6+39t^4+74t^2 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.90', '6.1310'] |
| 2-strand cable arrow polynomial of the knot is: -384*K1**2*K2**4 + 384*K1**2*K2**3 - 672*K1**2*K2**2 + 416*K1**2*K2 - 224*K1**2 + 256*K1*K2**3*K3 + 288*K1*K2*K3 - 192*K2**6 + 128*K2**4*K4 - 384*K2**4 - 32*K2**2*K3**2 - 16*K2**2*K4**2 + 192*K2**2*K4 + 144*K2**2 - 32*K3**2 - 16*K4**2 + 142 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.90'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.15', 'vk5.24', 'vk5.59', 'vk5.72', 'vk5.97', 'vk5.117', 'vk5.661', 'vk5.812', 'vk5.1191', 'vk5.1678'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is -. |
| The reverse -K is |
| The 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 | O1O2O3U4U5O4O5U1U2U3 |
| R3 orbit | {'O1O2O3U4U5O4O5U1U2U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U2U3O4O5U4U5 |
| Gauss code of K* | O1O2O3U1U2U3O4O5U4U5 |
| Gauss code of -K* | Same |
| Diagrammatic symmetry type | - |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -2 0 2 -1 1],[ 2 0 1 2 1 3],[ 0 -1 0 1 -1 1],[-2 -2 -1 0 -3 -1],[ 1 -1 1 3 0 1],[-1 -3 -1 1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -2],[-2 0 -1 -1 -3 -2],[-1 1 0 -1 -1 -3],[ 0 1 1 0 -1 -1],[ 1 3 1 1 0 -1],[ 2 2 3 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,2,1,1,3,2,1,1,3,1,1,1] |
| Phi over symmetry | [-2,-1,0,1,2,0,1,0,2,0,1,0,0,1,0] |
| Phi of -K | [-2,-1,0,1,2,0,1,0,2,0,1,0,0,1,0] |
| Phi of K* | [-2,-1,0,1,2,0,1,0,2,0,1,0,0,1,0] |
| Phi of -K* | [-2,-1,0,1,2,1,1,3,2,1,1,3,1,1,1] |
| Symmetry type of based matrix | - |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | -z-1 |
| Enhanced Jones-Krushkal polynomial | 8w^3z-9w^2z-w |
| Inner characteristic polynomial | t^5+29t^3+54t |
| Outer characteristic polynomial | t^6+39t^4+74t^2 |
| Flat arrow polynomial | -8*K1**3 + 4*K1*K2 + 4*K1 + 1 |
| 2-strand cable arrow polynomial | -384*K1**2*K2**4 + 384*K1**2*K2**3 - 672*K1**2*K2**2 + 416*K1**2*K2 - 224*K1**2 + 256*K1*K2**3*K3 + 288*K1*K2*K3 - 192*K2**6 + 128*K2**4*K4 - 384*K2**4 - 32*K2**2*K3**2 - 16*K2**2*K4**2 + 192*K2**2*K4 + 144*K2**2 - 32*K3**2 - 16*K4**2 + 142 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{4, 5}, {1, 3}, {2}]] |
| If K is slice | True |