| Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,1,2,2,3,2,1,2,3,1,1,2,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.702'] |
| Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.217', '6.219', '6.304', '6.349', '6.390', '6.400', '6.416', '6.515', '6.518', '6.530', '6.582', '6.616', '6.629', '6.641', '6.645', '6.702', '6.710', '6.715', '6.729', '6.733', '6.734', '6.802', '6.840', '6.845', '6.854', '6.860', '6.900', '6.905', '6.921', '6.924', '6.979', '6.980', '6.996', '6.1044', '6.1067', '6.1086', '6.1100', '6.1139', '6.1145', '6.1149', '6.1167', '6.1169', '6.1183', '6.1314'] |
| Outer characteristic polynomial of the knot is: t^7+59t^5+202t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.702'] |
| 2-strand cable arrow polynomial of the knot is: -144*K1**4 + 32*K1**3*K2*K3 - 672*K1**3*K3 - 480*K1**2*K2**2 - 32*K1**2*K2*K4 + 2976*K1**2*K2 - 176*K1**2*K3**2 - 3804*K1**2 - 160*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 3504*K1*K2*K3 + 648*K1*K3*K4 + 64*K1*K4*K5 - 72*K2**4 - 208*K2**2*K3**2 - 8*K2**2*K4**2 + 432*K2**2*K4 - 2726*K2**2 + 304*K2*K3*K5 + 8*K2*K4*K6 - 1696*K3**2 - 458*K4**2 - 124*K5**2 - 2*K6**2 + 2824 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.702'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81585', 'vk6.81667', 'vk6.81671', 'vk6.81904', 'vk6.81907', 'vk6.82102', 'vk6.82267', 'vk6.82273', 'vk6.82343', 'vk6.82348', 'vk6.82622', 'vk6.82626', 'vk6.82868', 'vk6.82876', 'vk6.83147', 'vk6.83154', 'vk6.83378', 'vk6.83387', 'vk6.84157', 'vk6.84659', 'vk6.84974', 'vk6.84977', 'vk6.85955', 'vk6.85966', 'vk6.86185', 'vk6.86189', 'vk6.86428', 'vk6.88131', 'vk6.89050', 'vk6.89054', 'vk6.89723', 'vk6.90035'] |
| 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 | O1O2O3O4U5O6U2U1O5U3U6U4 |
| R3 orbit | {'O1O2O3O4U5O6U2U1O5U3U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U5U2O6U4U3O5U6 |
| Gauss code of K* | O1O2O3U4U5U1U3O6U2O5O4U6 |
| Gauss code of -K* | O1O2O3U4O5O6U2O4U1U3U6U5 |
| 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 0 3 -1 2],[ 2 0 0 1 3 1 2],[ 2 0 0 0 2 2 1],[ 0 -1 0 0 2 0 0],[-3 -3 -2 -2 0 -2 -1],[ 1 -1 -2 0 2 0 2],[-2 -2 -1 0 1 -2 0]] |
| Primitive based matrix | [[ 0 3 2 0 -1 -2 -2],[-3 0 -1 -2 -2 -2 -3],[-2 1 0 0 -2 -1 -2],[ 0 2 0 0 0 0 -1],[ 1 2 2 0 0 -2 -1],[ 2 2 1 0 2 0 0],[ 2 3 2 1 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,1,2,2,1,2,2,2,3,0,2,1,2,0,0,1,2,1,0] |
| Phi over symmetry | [-3,-2,0,1,2,2,0,1,2,2,3,2,1,2,3,1,1,2,0,-1,0] |
| Phi of -K | [-2,-2,-1,0,2,3,0,-1,2,3,3,0,1,2,2,1,1,2,2,1,0] |
| Phi of K* | [-3,-2,0,1,2,2,0,1,2,2,3,2,1,2,3,1,1,2,0,-1,0] |
| Phi of -K* | [-2,-2,-1,0,2,3,0,1,1,2,3,2,0,1,2,0,2,2,0,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+5w^3z^2-6w^3z+22w^2z+21w |
| Inner characteristic polynomial | t^6+37t^4+109t^2+4 |
| Outer characteristic polynomial | t^7+59t^5+202t^3+11t |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -144*K1**4 + 32*K1**3*K2*K3 - 672*K1**3*K3 - 480*K1**2*K2**2 - 32*K1**2*K2*K4 + 2976*K1**2*K2 - 176*K1**2*K3**2 - 3804*K1**2 - 160*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 3504*K1*K2*K3 + 648*K1*K3*K4 + 64*K1*K4*K5 - 72*K2**4 - 208*K2**2*K3**2 - 8*K2**2*K4**2 + 432*K2**2*K4 - 2726*K2**2 + 304*K2*K3*K5 + 8*K2*K4*K6 - 1696*K3**2 - 458*K4**2 - 124*K5**2 - 2*K6**2 + 2824 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}]] |
| If K is slice | False |