| Min(phi) over symmetries of the knot is: [-3,-1,-1,1,1,3,-1,1,1,2,4,1,0,0,1,0,1,2,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.314'] |
| Arrow polynomial of the knot is: -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932'] |
| Outer characteristic polynomial of the knot is: t^7+81t^5+54t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.314'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 224*K1**4*K2 - 768*K1**4 + 448*K1**3*K2*K3 - 384*K1**3*K3 - 1024*K1**2*K2**4 + 1792*K1**2*K2**3 - 5184*K1**2*K2**2 - 192*K1**2*K2*K4 + 5016*K1**2*K2 - 704*K1**2*K3**2 - 32*K1**2*K3*K5 - 3456*K1**2 + 1536*K1*K2**3*K3 - 1120*K1*K2**2*K3 - 128*K1*K2**2*K5 + 64*K1*K2*K3**3 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5624*K1*K2*K3 + 784*K1*K3*K4 + 32*K1*K4*K5 - 1680*K2**4 - 896*K2**2*K3**2 - 16*K2**2*K4**2 + 1088*K2**2*K4 - 1844*K2**2 + 432*K2*K3*K5 + 16*K2*K4*K6 - 32*K3**4 + 16*K3**2*K6 - 1592*K3**2 - 284*K4**2 - 72*K5**2 - 4*K6**2 + 2722 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.314'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11509', 'vk6.11832', 'vk6.12847', 'vk6.13164', 'vk6.20275', 'vk6.21600', 'vk6.27543', 'vk6.29111', 'vk6.31274', 'vk6.31648', 'vk6.32420', 'vk6.32845', 'vk6.38946', 'vk6.41181', 'vk6.45715', 'vk6.47420', 'vk6.52272', 'vk6.52527', 'vk6.53101', 'vk6.53427', 'vk6.57108', 'vk6.58286', 'vk6.61691', 'vk6.62844', 'vk6.63791', 'vk6.63913', 'vk6.64227', 'vk6.64433', 'vk6.66739', 'vk6.67615', 'vk6.69391', 'vk6.70121'] |
| 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 | O1O2O3O4O5U2U3O6U5U4U1U6 |
| R3 orbit | {'O1O2O3O4O5U2U3O6U5U4U1U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U6U5U2U1O6U3U4 |
| Gauss code of K* | O1O2O3O4U3U5U6U2U1O5O6U4 |
| Gauss code of -K* | O1O2O3O4U1O5O6U4U3U5U6U2 |
| 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 -1 -3 -1 1 1 3],[ 1 0 -3 -1 2 2 3],[ 3 3 0 1 3 2 2],[ 1 1 -1 0 2 1 2],[-1 -2 -3 -2 0 0 2],[-1 -2 -2 -1 0 0 1],[-3 -3 -2 -2 -2 -1 0]] |
| Primitive based matrix | [[ 0 3 1 1 -1 -1 -3],[-3 0 -1 -2 -2 -3 -2],[-1 1 0 0 -1 -2 -2],[-1 2 0 0 -2 -2 -3],[ 1 2 1 2 0 1 -1],[ 1 3 2 2 -1 0 -3],[ 3 2 2 3 1 3 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,1,1,3,1,2,2,3,2,0,1,2,2,2,2,3,-1,1,3] |
| Phi over symmetry | [-3,-1,-1,1,1,3,-1,1,1,2,4,1,0,0,1,0,1,2,0,0,1] |
| Phi of -K | [-3,-1,-1,1,1,3,-1,1,1,2,4,1,0,0,1,0,1,2,0,0,1] |
| Phi of K* | [-3,-1,-1,1,1,3,0,1,1,2,4,0,0,0,1,0,1,2,-1,-1,1] |
| Phi of -K* | [-3,-1,-1,1,1,3,1,3,2,3,2,1,1,2,2,2,2,3,0,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 6z^2+23z+23 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+8w^3z^2-2w^3z+25w^2z+23w |
| Inner characteristic polynomial | t^6+59t^4+24t^2 |
| Outer characteristic polynomial | t^7+81t^5+54t^3+8t |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 224*K1**4*K2 - 768*K1**4 + 448*K1**3*K2*K3 - 384*K1**3*K3 - 1024*K1**2*K2**4 + 1792*K1**2*K2**3 - 5184*K1**2*K2**2 - 192*K1**2*K2*K4 + 5016*K1**2*K2 - 704*K1**2*K3**2 - 32*K1**2*K3*K5 - 3456*K1**2 + 1536*K1*K2**3*K3 - 1120*K1*K2**2*K3 - 128*K1*K2**2*K5 + 64*K1*K2*K3**3 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5624*K1*K2*K3 + 784*K1*K3*K4 + 32*K1*K4*K5 - 1680*K2**4 - 896*K2**2*K3**2 - 16*K2**2*K4**2 + 1088*K2**2*K4 - 1844*K2**2 + 432*K2*K3*K5 + 16*K2*K4*K6 - 32*K3**4 + 16*K3**2*K6 - 1592*K3**2 - 284*K4**2 - 72*K5**2 - 4*K6**2 + 2722 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{5, 6}, {3, 4}, {1, 2}]] |
| If K is slice | False |