| Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,-1,2,3,2,3,2,2,1,1,1,1,1,1,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.582'] |
| 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+63t^5+40t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.582'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 64*K1**4*K2 - 784*K1**4 + 32*K1**3*K2*K3 + 32*K1**3*K3*K4 - 288*K1**2*K2**2 + 936*K1**2*K2 - 400*K1**2*K3**2 - 128*K1**2*K4**2 - 672*K1**2 + 1096*K1*K2*K3 + 760*K1*K3*K4 + 88*K1*K4*K5 - 8*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 48*K2**2*K4 - 654*K2**2 + 32*K2*K3*K5 + 8*K2*K4*K6 - 660*K3**2 - 298*K4**2 - 28*K5**2 - 2*K6**2 + 912 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.582'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16579', 'vk6.16672', 'vk6.18129', 'vk6.18463', 'vk6.22982', 'vk6.23103', 'vk6.24588', 'vk6.24999', 'vk6.34971', 'vk6.35092', 'vk6.35394', 'vk6.35815', 'vk6.36719', 'vk6.37136', 'vk6.39404', 'vk6.41595', 'vk6.42544', 'vk6.42655', 'vk6.42871', 'vk6.43150', 'vk6.43991', 'vk6.44301', 'vk6.45984', 'vk6.47658', 'vk6.54810', 'vk6.55362', 'vk6.56241', 'vk6.57402', 'vk6.59242', 'vk6.59803', 'vk6.60845', 'vk6.62073', 'vk6.64784', 'vk6.64849', 'vk6.65600', 'vk6.65905', 'vk6.68086', 'vk6.68151', 'vk6.68675', 'vk6.68884'] |
| 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 | O1O2O3O4U1O5U4O6U2U6U5U3 |
| R3 orbit | {'O1O2O3O4U1O5U4O6U2U6U5U3', 'O1O2O3O4U1U3O5O6U2U6U4U5'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U2U5U6U3O6U1O5U4 |
| Gauss code of K* | O1O2O3O4U5U1U4U6O5U3O6U2 |
| Gauss code of -K* | O1O2O3O4U3O5U2O6U5U1U4U6 |
| 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 -3 -2 2 0 2 1],[ 3 0 2 3 1 2 1],[ 2 -2 0 3 0 3 1],[-2 -3 -3 0 -1 1 0],[ 0 -1 0 1 0 1 0],[-2 -2 -3 -1 -1 0 0],[-1 -1 -1 0 0 0 0]] |
| Primitive based matrix | [[ 0 2 2 1 0 -2 -3],[-2 0 1 0 -1 -3 -3],[-2 -1 0 0 -1 -3 -2],[-1 0 0 0 0 -1 -1],[ 0 1 1 0 0 0 -1],[ 2 3 3 1 0 0 -2],[ 3 3 2 1 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,0,2,3,-1,0,1,3,3,0,1,3,2,0,1,1,0,1,2] |
| Phi over symmetry | [-3,-2,0,1,2,2,-1,2,3,2,3,2,2,1,1,1,1,1,1,1,-1] |
| Phi of -K | [-3,-2,0,1,2,2,-1,2,3,2,3,2,2,1,1,1,1,1,1,1,-1] |
| Phi of K* | [-2,-2,-1,0,2,3,-1,1,1,1,3,1,1,1,2,1,2,3,2,2,-1] |
| Phi of -K* | [-3,-2,0,1,2,2,2,1,1,2,3,0,1,3,3,0,1,1,0,0,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 11z+23 |
| Enhanced Jones-Krushkal polynomial | 11w^2z+23w |
| Inner characteristic polynomial | t^6+41t^4+11t^2 |
| Outer characteristic polynomial | t^7+63t^5+40t^3 |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -64*K1**6 + 64*K1**4*K2 - 784*K1**4 + 32*K1**3*K2*K3 + 32*K1**3*K3*K4 - 288*K1**2*K2**2 + 936*K1**2*K2 - 400*K1**2*K3**2 - 128*K1**2*K4**2 - 672*K1**2 + 1096*K1*K2*K3 + 760*K1*K3*K4 + 88*K1*K4*K5 - 8*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 48*K2**2*K4 - 654*K2**2 + 32*K2*K3*K5 + 8*K2*K4*K6 - 660*K3**2 - 298*K4**2 - 28*K5**2 - 2*K6**2 + 912 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}]] |
| If K is slice | False |