| Min(phi) over symmetries of the knot is: [-4,-2,1,1,2,2,1,1,3,2,4,1,2,1,3,-1,0,-1,0,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.439'] |
| Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + K4 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.71', '6.148', '6.170', '6.253', '6.259', '6.298', '6.439', '6.453', '6.499', '6.503'] |
| Outer characteristic polynomial of the knot is: t^7+93t^5+241t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.439'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 96*K1**4*K2 - 176*K1**4 + 128*K1**3*K2**3*K3 + 416*K1**3*K2*K3 - 320*K1**2*K2**4 + 160*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 1424*K1**2*K2**2 + 752*K1**2*K2 - 272*K1**2*K3**2 - 416*K1**2 + 416*K1*K2**3*K3 + 32*K1*K2*K3**3 + 1344*K1*K2*K3 + 224*K1*K3*K4 + 40*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 512*K2**4 + 32*K2**3*K3*K5 - 272*K2**2*K3**2 - 80*K2**2*K4**2 + 408*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 386*K2**2 + 232*K2*K3*K5 + 32*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 + 8*K3**2*K6 - 492*K3**2 - 226*K4**2 - 104*K5**2 - 14*K6**2 - 4*K7**2 - 2*K8**2 + 714 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.439'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16912', 'vk6.17156', 'vk6.17495', 'vk6.17506', 'vk6.17550', 'vk6.17563', 'vk6.21891', 'vk6.24019', 'vk6.24034', 'vk6.24103', 'vk6.27947', 'vk6.29428', 'vk6.35326', 'vk6.35760', 'vk6.36277', 'vk6.36292', 'vk6.36359', 'vk6.39361', 'vk6.41539', 'vk6.43432', 'vk6.43443', 'vk6.43472', 'vk6.45926', 'vk6.47615', 'vk6.55071', 'vk6.55322', 'vk6.55617', 'vk6.55620', 'vk6.55649', 'vk6.58542', 'vk6.60131', 'vk6.60134', 'vk6.60166', 'vk6.63030', 'vk6.64908', 'vk6.65123', 'vk6.65324', 'vk6.65359', 'vk6.68496', 'vk6.68520'] |
| 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 | O1O2O3O4O5U6U2O6U3U1U4U5 |
| R3 orbit | {'O1O2O3O4O5U6U2O6U3U1U4U5', 'O1O2O3O4O5U3U6U2O6U1U4U5'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U1U2U5U3O6U4U6 |
| Gauss code of K* | O1O2O3O4U2U5U1U3U4O6O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6O5U1U2U4U6U3 |
| 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 -2 -2 -1 2 4 -1],[ 2 0 0 1 3 4 1],[ 2 0 0 0 1 2 2],[ 1 -1 0 0 1 2 1],[-2 -3 -1 -1 0 1 -2],[-4 -4 -2 -2 -1 0 -4],[ 1 -1 -2 -1 2 4 0]] |
| Primitive based matrix | [[ 0 4 2 -1 -1 -2 -2],[-4 0 -1 -2 -4 -2 -4],[-2 1 0 -1 -2 -1 -3],[ 1 2 1 0 1 0 -1],[ 1 4 2 -1 0 -2 -1],[ 2 2 1 0 2 0 0],[ 2 4 3 1 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-4,-2,1,1,2,2,1,2,4,2,4,1,2,1,3,-1,0,1,2,1,0] |
| Phi over symmetry | [-4,-2,1,1,2,2,1,1,3,2,4,1,2,1,3,-1,0,-1,0,1,0] |
| Phi of -K | [-2,-2,-1,-1,2,4,0,-1,1,3,4,0,0,1,2,1,1,1,2,3,1] |
| Phi of K* | [-4,-2,1,1,2,2,1,1,3,2,4,1,2,1,3,-1,0,-1,0,1,0] |
| Phi of -K* | [-2,-2,-1,-1,2,4,0,0,2,1,2,1,1,3,4,1,1,2,2,4,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^4+t^2+2t |
| Normalized Jones-Krushkal polynomial | 3z+7 |
| Enhanced Jones-Krushkal polynomial | -8w^3z+11w^2z+7w |
| Inner characteristic polynomial | t^6+63t^4+146t^2 |
| Outer characteristic polynomial | t^7+93t^5+241t^3 |
| Flat arrow polynomial | 4*K1**3 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + K4 + 1 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 96*K1**4*K2 - 176*K1**4 + 128*K1**3*K2**3*K3 + 416*K1**3*K2*K3 - 320*K1**2*K2**4 + 160*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 1424*K1**2*K2**2 + 752*K1**2*K2 - 272*K1**2*K3**2 - 416*K1**2 + 416*K1*K2**3*K3 + 32*K1*K2*K3**3 + 1344*K1*K2*K3 + 224*K1*K3*K4 + 40*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 512*K2**4 + 32*K2**3*K3*K5 - 272*K2**2*K3**2 - 80*K2**2*K4**2 + 408*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 386*K2**2 + 232*K2*K3*K5 + 32*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 + 8*K3**2*K6 - 492*K3**2 - 226*K4**2 - 104*K5**2 - 14*K6**2 - 4*K7**2 - 2*K8**2 + 714 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {2, 5}, {1, 3}]] |
| If K is slice | False |