| Min(phi) over symmetries of the knot is: [-3,-3,0,1,2,3,0,0,2,3,2,1,3,3,3,1,0,1,0,1,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.304'] |
| 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+94t^5+116t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.304'] |
| 2-strand cable arrow polynomial of the knot is: -144*K1**4 + 32*K1**3*K2*K3 - 64*K1**3*K3 - 896*K1**2*K2**2 - 32*K1**2*K2*K4 + 1936*K1**2*K2 - 16*K1**2*K3**2 - 1820*K1**2 - 608*K1*K2**2*K3 + 2144*K1*K2*K3 + 400*K1*K3*K4 + 8*K1*K5*K6 - 72*K2**4 - 272*K2**2*K3**2 - 72*K2**2*K4**2 + 592*K2**2*K4 - 1662*K2**2 + 200*K2*K3*K5 + 64*K2*K4*K6 - 832*K3**2 - 314*K4**2 - 36*K5**2 - 18*K6**2 + 1480 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.304'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73323', 'vk6.73465', 'vk6.73737', 'vk6.73854', 'vk6.74570', 'vk6.74837', 'vk6.75217', 'vk6.75646', 'vk6.76048', 'vk6.76397', 'vk6.76772', 'vk6.76888', 'vk6.78212', 'vk6.78661', 'vk6.79003', 'vk6.79243', 'vk6.79570', 'vk6.79725', 'vk6.80025', 'vk6.80283', 'vk6.80537', 'vk6.80739', 'vk6.80995', 'vk6.81083', 'vk6.81619', 'vk6.81810', 'vk6.82355', 'vk6.82373', 'vk6.82733', 'vk6.84231', 'vk6.84300', 'vk6.84325', 'vk6.84360', 'vk6.84420', 'vk6.84505', 'vk6.85241', 'vk6.86763', 'vk6.87573', 'vk6.88267', 'vk6.88704'] |
| 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 | O1O2O3O4O5U2U1O6U5U3U6U4 |
| R3 orbit | {'O1O2O3O4O5U2U1U4O6U3U5U6', 'O1O2O3O4O5U2U1O6U5U3U6U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U2U6U3U1O6U5U4 |
| Gauss code of K* | O1O2O3O4U5U6U2U4U1O6O5U3 |
| Gauss code of -K* | O1O2O3O4U2O5O6U4U1U3U6U5 |
| 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 1 2],[ 3 0 0 3 4 2 2],[ 3 0 0 2 3 1 2],[ 0 -3 -2 0 2 0 2],[-3 -4 -3 -2 0 -1 1],[-1 -2 -1 0 1 0 1],[-2 -2 -2 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 2 1 0 -3 -3],[-3 0 1 -1 -2 -3 -4],[-2 -1 0 -1 -2 -2 -2],[-1 1 1 0 0 -1 -2],[ 0 2 2 0 0 -2 -3],[ 3 3 2 1 2 0 0],[ 3 4 2 2 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,0,3,3,-1,1,2,3,4,1,2,2,2,0,1,2,2,3,0] |
| Phi over symmetry | [-3,-3,0,1,2,3,0,0,2,3,2,1,3,3,3,1,0,1,0,1,2] |
| Phi of -K | [-3,-3,0,1,2,3,0,0,2,3,2,1,3,3,3,1,0,1,0,1,2] |
| Phi of K* | [-3,-2,-1,0,3,3,2,1,1,2,3,0,0,3,3,1,2,3,0,1,0] |
| Phi of -K* | [-3,-3,0,1,2,3,0,2,1,2,3,3,2,2,4,0,2,2,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+17w^2z+19w |
| Inner characteristic polynomial | t^6+62t^4+21t^2 |
| Outer characteristic polynomial | t^7+94t^5+116t^3+3t |
| 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 - 64*K1**3*K3 - 896*K1**2*K2**2 - 32*K1**2*K2*K4 + 1936*K1**2*K2 - 16*K1**2*K3**2 - 1820*K1**2 - 608*K1*K2**2*K3 + 2144*K1*K2*K3 + 400*K1*K3*K4 + 8*K1*K5*K6 - 72*K2**4 - 272*K2**2*K3**2 - 72*K2**2*K4**2 + 592*K2**2*K4 - 1662*K2**2 + 200*K2*K3*K5 + 64*K2*K4*K6 - 832*K3**2 - 314*K4**2 - 36*K5**2 - 18*K6**2 + 1480 |
| 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}], [{6}, {5}, {4}, {2, 3}, {1}], [{6}, {5}, {4}, {3}, {1, 2}]] |
| If K is slice | False |