| Min(phi) over symmetries of the knot is: [-3,-3,0,1,2,3,0,0,3,3,2,1,3,4,3,0,2,1,1,2,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.518'] |
| 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+82t^5+164t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.518'] |
| 2-strand cable arrow polynomial of the knot is: -2304*K1**2*K2**2 + 2632*K1**2*K2 - 2568*K1**2 + 320*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 352*K1*K2**2*K3 - 32*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4080*K1*K2*K3 - 96*K1*K2*K4*K5 + 488*K1*K3*K4 + 96*K1*K4*K5 + 24*K1*K5*K6 - 1336*K2**4 - 1632*K2**2*K3**2 - 168*K2**2*K4**2 + 1352*K2**2*K4 - 1942*K2**2 - 128*K2*K3**2*K4 + 1392*K2*K3*K5 + 232*K2*K4*K6 + 64*K3**2*K6 - 1568*K3**2 - 530*K4**2 - 320*K5**2 - 74*K6**2 + 2464 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.518'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16369', 'vk6.16412', 'vk6.18109', 'vk6.18445', 'vk6.22703', 'vk6.22806', 'vk6.24558', 'vk6.24975', 'vk6.34676', 'vk6.34757', 'vk6.36699', 'vk6.37120', 'vk6.42329', 'vk6.42375', 'vk6.43975', 'vk6.44290', 'vk6.54628', 'vk6.54655', 'vk6.55927', 'vk6.56221', 'vk6.59107', 'vk6.59176', 'vk6.60457', 'vk6.60820', 'vk6.64654', 'vk6.64704', 'vk6.65579', 'vk6.65890', 'vk6.68004', 'vk6.68032', 'vk6.68656', 'vk6.68869'] |
| 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 | O1O2O3O4O5U2U1U5O6U3U6U4 |
| R3 orbit | {'O1O2O3O4O5U2U1U5O6U3U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U2U6U3O6U1U5U4 |
| Gauss code of K* | O1O2O3U4O5O4O6U2U1U5U6U3 |
| Gauss code of -K* | O1O2O3U2O4O5O6U4U1U3U6U5 |
| 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 -3 -3 0 3 2 1],[ 3 0 0 3 4 2 1],[ 3 0 0 2 3 1 1],[ 0 -3 -2 0 2 0 1],[-3 -4 -3 -2 0 0 0],[-2 -2 -1 0 0 0 0],[-1 -1 -1 -1 0 0 0]] |
| Primitive based matrix | [[ 0 3 2 1 0 -3 -3],[-3 0 0 0 -2 -3 -4],[-2 0 0 0 0 -1 -2],[-1 0 0 0 -1 -1 -1],[ 0 2 0 1 0 -2 -3],[ 3 3 1 1 2 0 0],[ 3 4 2 1 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,0,3,3,0,0,2,3,4,0,0,1,2,1,1,1,2,3,0] |
| Phi over symmetry | [-3,-3,0,1,2,3,0,0,3,3,2,1,3,4,3,0,2,1,1,2,1] |
| Phi of -K | [-3,-3,0,1,2,3,0,0,3,3,2,1,3,4,3,0,2,1,1,2,1] |
| Phi of K* | [-3,-2,-1,0,3,3,1,2,1,2,3,1,2,3,4,0,3,3,0,1,0] |
| Phi of -K* | [-3,-3,0,1,2,3,0,2,1,1,3,3,1,2,4,1,0,2,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 8z^2+25z+19 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+10w^3z^2-2w^3z+27w^2z+19w |
| Inner characteristic polynomial | t^6+50t^4+39t^2 |
| Outer characteristic polynomial | t^7+82t^5+164t^3+8t |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -2304*K1**2*K2**2 + 2632*K1**2*K2 - 2568*K1**2 + 320*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 352*K1*K2**2*K3 - 32*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4080*K1*K2*K3 - 96*K1*K2*K4*K5 + 488*K1*K3*K4 + 96*K1*K4*K5 + 24*K1*K5*K6 - 1336*K2**4 - 1632*K2**2*K3**2 - 168*K2**2*K4**2 + 1352*K2**2*K4 - 1942*K2**2 - 128*K2*K3**2*K4 + 1392*K2*K3*K5 + 232*K2*K4*K6 + 64*K3**2*K6 - 1568*K3**2 - 530*K4**2 - 320*K5**2 - 74*K6**2 + 2464 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{5, 6}, {4}, {3}, {2}, {1}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {1, 3}, {2}]] |
| If K is slice | False |