| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,1,2,1,1,0,1,0,1,1,0,2,1,2,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1342'] |
| Arrow polynomial of the knot is: -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.235', '6.379', '6.411', '6.547', '6.811', '6.818', '6.823', '6.897', '6.898', '6.1008', '6.1053', '6.1109', '6.1110', '6.1130', '6.1222', '6.1239', '6.1303', '6.1307', '6.1342', '6.1491', '6.1495', '6.1496', '6.1519', '6.1592', '6.1593', '6.1642', '6.1652', '6.1653', '6.1671', '6.1673', '6.1717'] |
| Outer characteristic polynomial of the knot is: t^7+29t^5+49t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1342'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**6 + 736*K1**4*K2 - 4144*K1**4 + 480*K1**3*K2*K3 + 128*K1**3*K3*K4 - 768*K1**3*K3 - 3328*K1**2*K2**2 - 576*K1**2*K2*K4 + 9248*K1**2*K2 - 1232*K1**2*K3**2 - 128*K1**2*K3*K5 - 176*K1**2*K4**2 - 6020*K1**2 - 416*K1*K2**2*K3 - 160*K1*K2*K3*K4 + 6664*K1*K2*K3 + 2016*K1*K3*K4 + 392*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**4 - 16*K2**2*K3**2 - 16*K2**2*K4**2 + 784*K2**2*K4 - 5204*K2**2 + 264*K2*K3*K5 + 32*K2*K4*K6 + 32*K3**2*K6 - 2660*K3**2 - 948*K4**2 - 240*K5**2 - 44*K6**2 + 5458 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1342'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3655', 'vk6.3752', 'vk6.3945', 'vk6.4042', 'vk6.4481', 'vk6.4578', 'vk6.5863', 'vk6.5992', 'vk6.7148', 'vk6.7325', 'vk6.7418', 'vk6.7920', 'vk6.8041', 'vk6.9350', 'vk6.17900', 'vk6.17997', 'vk6.18762', 'vk6.24439', 'vk6.24881', 'vk6.25344', 'vk6.37501', 'vk6.43874', 'vk6.44228', 'vk6.44533', 'vk6.48295', 'vk6.48360', 'vk6.50086', 'vk6.50200', 'vk6.50565', 'vk6.50630', 'vk6.55861', 'vk6.60736'] |
| 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 | O1O2O3U1O4O5U4U3U5O6U2U6 |
| R3 orbit | {'O1O2O3U1O4O5U4U3U5O6U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U2O4U5U1U6O5O6U3 |
| Gauss code of K* | O1O2O3U4O5O4U6U5U2O6U1U3 |
| Gauss code of -K* | O1O2O3U1U3O4U2U5U4O6O5U6 |
| 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 0 0 -1 2 1],[ 2 0 2 1 0 1 1],[ 0 -2 0 -1 -1 2 1],[ 0 -1 1 0 0 2 0],[ 1 0 1 0 0 1 0],[-2 -1 -2 -2 -1 0 0],[-1 -1 -1 0 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 -2 -2 -1 -1],[-1 0 0 0 -1 0 -1],[ 0 2 0 0 1 0 -1],[ 0 2 1 -1 0 -1 -2],[ 1 1 0 0 1 0 0],[ 2 1 1 1 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,2,2,1,1,0,1,0,1,-1,0,1,1,2,0] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,1,2,1,1,0,1,0,1,1,0,2,1,2,0] |
| Phi of -K | [-2,-1,0,0,1,2,1,0,1,2,3,0,1,2,2,1,0,0,1,0,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,0,0,2,3,0,1,2,2,-1,0,0,1,1,1] |
| Phi of -K* | [-2,-1,0,0,1,2,0,1,2,1,1,0,1,0,1,1,0,2,1,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 19z+39 |
| Enhanced Jones-Krushkal polynomial | 19w^2z+39w |
| Inner characteristic polynomial | t^6+19t^4+21t^2 |
| Outer characteristic polynomial | t^7+29t^5+49t^3+5t |
| Flat arrow polynomial | -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5 |
| 2-strand cable arrow polynomial | -128*K1**6 + 736*K1**4*K2 - 4144*K1**4 + 480*K1**3*K2*K3 + 128*K1**3*K3*K4 - 768*K1**3*K3 - 3328*K1**2*K2**2 - 576*K1**2*K2*K4 + 9248*K1**2*K2 - 1232*K1**2*K3**2 - 128*K1**2*K3*K5 - 176*K1**2*K4**2 - 6020*K1**2 - 416*K1*K2**2*K3 - 160*K1*K2*K3*K4 + 6664*K1*K2*K3 + 2016*K1*K3*K4 + 392*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**4 - 16*K2**2*K3**2 - 16*K2**2*K4**2 + 784*K2**2*K4 - 5204*K2**2 + 264*K2*K3*K5 + 32*K2*K4*K6 + 32*K3**2*K6 - 2660*K3**2 - 948*K4**2 - 240*K5**2 - 44*K6**2 + 5458 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]] |
| If K is slice | False |