| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,1,3,3,0,0,1,1,0,0,0,-1,-1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.390'] |
| 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+56t^5+21t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.390'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**6 - 128*K1**4*K2**2 + 288*K1**4*K2 - 224*K1**4 + 64*K1**3*K2*K3 - 192*K1**2*K2**2 + 304*K1**2*K2 - 116*K1**2 + 168*K1*K2*K3 + 40*K1*K3*K4 + 8*K1*K4*K5 - 24*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 48*K2**2*K4 - 174*K2**2 + 24*K2*K3*K5 + 8*K2*K4*K6 - 80*K3**2 - 42*K4**2 - 12*K5**2 - 2*K6**2 + 192 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.390'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11057', 'vk6.11135', 'vk6.11563', 'vk6.11902', 'vk6.12219', 'vk6.12326', 'vk6.13216', 'vk6.19239', 'vk6.19331', 'vk6.19534', 'vk6.19624', 'vk6.19987', 'vk6.21159', 'vk6.22395', 'vk6.22715', 'vk6.26049', 'vk6.26095', 'vk6.26517', 'vk6.26876', 'vk6.27014', 'vk6.28431', 'vk6.28638', 'vk6.30626', 'vk6.30721', 'vk6.31352', 'vk6.31364', 'vk6.31762', 'vk6.31928', 'vk6.32526', 'vk6.32925', 'vk6.34759', 'vk6.38113', 'vk6.38422', 'vk6.38873', 'vk6.40142', 'vk6.40157', 'vk6.40428', 'vk6.42923', 'vk6.44638', 'vk6.44749', 'vk6.45167', 'vk6.45306', 'vk6.45638', 'vk6.46663', 'vk6.52336', 'vk6.52598', 'vk6.52810', 'vk6.55190', 'vk6.56793', 'vk6.58215', 'vk6.59569', 'vk6.61293', 'vk6.62386', 'vk6.66278', 'vk6.66501', 'vk6.69151'] |
| 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 | O1O2O3O4O5U4U2O6U5U3U1U6 |
| R3 orbit | {'O1O2O3O4U3O5U2O6U4U5U1U6', 'O1O2O3O4O5U4U2O6U5U3U1U6', 'O1O2O3U2O4U1O5O6U3U4U5U6', 'O1O2O3O4U3U1O5O6U4U2U5U6'} |
| R3 orbit length | 4 |
| Gauss code of -K | O1O2O3O4O5U6U5U3U1O6U4U2 |
| Gauss code of K* | O1O2O3O4U3U5U2U6U1O6O5U4 |
| Gauss code of -K* | O1O2O3O4U1O5O6U4U6U3U5U2 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -1 -2 0 -1 1 3],[ 1 0 -2 1 -1 2 3],[ 2 2 0 2 0 2 2],[ 0 -1 -2 0 -1 1 2],[ 1 1 0 1 0 1 1],[-1 -2 -2 -1 -1 0 1],[-3 -3 -2 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 -2 -1 -3 -2],[-1 1 0 -1 -1 -2 -2],[ 0 2 1 0 -1 -1 -2],[ 1 1 1 1 0 1 0],[ 1 3 2 1 -1 0 -2],[ 2 2 2 2 0 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,1,2,1,3,2,1,1,2,2,1,1,2,-1,0,2] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,1,1,3,3,0,0,1,1,0,0,0,-1,-1,1] |
| Phi of -K | [-2,-1,-1,0,1,3,-1,1,0,1,3,1,0,0,1,0,1,3,0,1,1] |
| Phi of K* | [-3,-1,0,1,1,2,1,1,1,3,3,0,0,1,1,0,0,0,-1,-1,1] |
| Phi of -K* | [-2,-1,-1,0,1,3,0,2,2,2,2,1,1,1,1,1,2,3,1,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 5z+11 |
| Enhanced Jones-Krushkal polynomial | 5w^2z+11w |
| Inner characteristic polynomial | t^6+40t^4 |
| Outer characteristic polynomial | t^7+56t^5+21t^3 |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -128*K1**6 - 128*K1**4*K2**2 + 288*K1**4*K2 - 224*K1**4 + 64*K1**3*K2*K3 - 192*K1**2*K2**2 + 304*K1**2*K2 - 116*K1**2 + 168*K1*K2*K3 + 40*K1*K3*K4 + 8*K1*K4*K5 - 24*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 48*K2**2*K4 - 174*K2**2 + 24*K2*K3*K5 + 8*K2*K4*K6 - 80*K3**2 - 42*K4**2 - 12*K5**2 - 2*K6**2 + 192 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {2, 5}, {3, 4}, {1}]] |
| If K is slice | False |