| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,0,2,3,3,0,1,2,2,0,1,0,0,1,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.801'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355'] |
| Outer characteristic polynomial of the knot is: t^7+54t^5+63t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.801'] |
| 2-strand cable arrow polynomial of the knot is: -960*K1**4*K2**2 + 2112*K1**4*K2 - 4512*K1**4 - 256*K1**3*K2**2*K3 + 416*K1**3*K2*K3 - 512*K1**3*K3 - 320*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2368*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 8768*K1**2*K2**2 - 736*K1**2*K2*K4 + 8744*K1**2*K2 - 320*K1**2*K3**2 - 2884*K1**2 + 1664*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 832*K1*K2**2*K3 - 448*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 6104*K1*K2*K3 + 544*K1*K3*K4 - 32*K2**6 + 160*K2**4*K4 - 1544*K2**4 - 656*K2**2*K3**2 - 104*K2**2*K4**2 + 880*K2**2*K4 - 2160*K2**2 + 192*K2*K3*K5 - 1100*K3**2 - 206*K4**2 + 3004 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.801'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19948', 'vk6.20050', 'vk6.21193', 'vk6.21336', 'vk6.26917', 'vk6.27111', 'vk6.28671', 'vk6.28804', 'vk6.38337', 'vk6.38500', 'vk6.40477', 'vk6.40705', 'vk6.45206', 'vk6.45396', 'vk6.47029', 'vk6.47148', 'vk6.56744', 'vk6.56850', 'vk6.57845', 'vk6.57993', 'vk6.61181', 'vk6.61373', 'vk6.62419', 'vk6.62540', 'vk6.66444', 'vk6.66563', 'vk6.67214', 'vk6.67358', 'vk6.69092', 'vk6.69211', 'vk6.69873', 'vk6.69956'] |
| 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 | O1O2O3O4U5O6U4U2U3O5U1U6 |
| R3 orbit | {'O1O2O3O4U5O6U4U2U3O5U1U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U4O6U2U3U1O5U6 |
| Gauss code of K* | O1O2O3U4O5O6U5U2U3U1O4U6 |
| Gauss code of -K* | O1O2O3U4O5U3U1U2U6O4O6U5 |
| 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 -1 -1 1 0 -2 3],[ 1 0 0 2 1 -2 3],[ 1 0 0 1 0 -1 2],[-1 -2 -1 0 0 -2 1],[ 0 -1 0 0 0 0 0],[ 2 2 1 2 0 0 3],[-3 -3 -2 -1 0 -3 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 0 -2 -3 -3],[-1 1 0 0 -1 -2 -2],[ 0 0 0 0 0 -1 0],[ 1 2 1 0 0 0 -1],[ 1 3 2 1 0 0 -2],[ 2 3 2 0 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,1,0,2,3,3,0,1,2,2,0,1,0,0,1,2] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,0,2,3,3,0,1,2,2,0,1,0,0,1,2] |
| Phi of -K | [-2,-1,-1,0,1,3,-1,0,2,1,2,0,0,0,1,1,1,2,1,3,1] |
| Phi of K* | [-3,-1,0,1,1,2,1,3,1,2,2,1,0,1,1,0,1,2,0,-1,0] |
| Phi of -K* | [-2,-1,-1,0,1,3,1,2,0,2,3,0,0,1,2,1,2,3,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 6z^2+27z+31 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+27w^2z+31w |
| Inner characteristic polynomial | t^6+38t^4+26t^2 |
| Outer characteristic polynomial | t^7+54t^5+63t^3+7t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -960*K1**4*K2**2 + 2112*K1**4*K2 - 4512*K1**4 - 256*K1**3*K2**2*K3 + 416*K1**3*K2*K3 - 512*K1**3*K3 - 320*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2368*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 8768*K1**2*K2**2 - 736*K1**2*K2*K4 + 8744*K1**2*K2 - 320*K1**2*K3**2 - 2884*K1**2 + 1664*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 832*K1*K2**2*K3 - 448*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 6104*K1*K2*K3 + 544*K1*K3*K4 - 32*K2**6 + 160*K2**4*K4 - 1544*K2**4 - 656*K2**2*K3**2 - 104*K2**2*K4**2 + 880*K2**2*K4 - 2160*K2**2 + 192*K2*K3*K5 - 1100*K3**2 - 206*K4**2 + 3004 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | False |