| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,1,3,3,0,1,1,2,1,0,2,0,-1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1297'] |
| 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+50t^5+118t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1297'] |
| 2-strand cable arrow polynomial of the knot is: -1600*K1**2*K2**4 + 3360*K1**2*K2**3 - 6896*K1**2*K2**2 - 736*K1**2*K2*K4 + 4584*K1**2*K2 - 3112*K1**2 + 1984*K1*K2**3*K3 - 992*K1*K2**2*K3 - 224*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4752*K1*K2*K3 + 456*K1*K3*K4 + 8*K1*K4*K5 - 288*K2**6 + 160*K2**4*K4 - 2616*K2**4 - 816*K2**2*K3**2 - 8*K2**2*K4**2 + 1648*K2**2*K4 - 720*K2**2 + 328*K2*K3*K5 - 968*K3**2 - 326*K4**2 - 32*K5**2 + 2148 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1297'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16787', 'vk6.16794', 'vk6.16818', 'vk6.16827', 'vk6.18157', 'vk6.18168', 'vk6.18491', 'vk6.18503', 'vk6.23199', 'vk6.23206', 'vk6.24616', 'vk6.25027', 'vk6.25042', 'vk6.35218', 'vk6.35247', 'vk6.36752', 'vk6.37171', 'vk6.37193', 'vk6.42695', 'vk6.42706', 'vk6.44333', 'vk6.44344', 'vk6.54980', 'vk6.55013', 'vk6.55964', 'vk6.55966', 'vk6.59368', 'vk6.59379', 'vk6.60502', 'vk6.65629', 'vk6.68169', 'vk6.68176'] |
| 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 | O1O2O3U2O4O5U6U3O6U1U4U5 |
| R3 orbit | {'O1O2O3U2O4O5U6U3O6U1U4U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5U3O6U1U6O4O5U2 |
| Gauss code of K* | O1O2O3U1U4U5O4U2U3O6O5U6 |
| Gauss code of -K* | O1O2O3U4O5O4U1U2O6U5U6U3 |
| 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 -1 0 1 3 -1],[ 2 0 -1 2 2 3 1],[ 1 1 0 1 1 1 0],[ 0 -2 -1 0 0 1 0],[-1 -2 -1 0 0 1 -1],[-3 -3 -1 -1 -1 0 -3],[ 1 -1 0 0 1 3 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 -1 -1 -3 -3],[-1 1 0 0 -1 -1 -2],[ 0 1 0 0 -1 0 -2],[ 1 1 1 1 0 0 1],[ 1 3 1 0 0 0 -1],[ 2 3 2 2 -1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,1,1,1,3,3,0,1,1,2,1,0,2,0,-1,1] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,1,1,3,3,0,1,1,2,1,0,2,0,-1,1] |
| Phi of -K | [-2,-1,-1,0,1,3,0,2,0,1,2,0,1,1,1,0,1,3,1,2,1] |
| Phi of K* | [-3,-1,0,1,1,2,1,2,1,3,2,1,1,1,1,1,0,0,0,0,2] |
| Phi of -K* | [-2,-1,-1,0,1,3,-1,1,2,2,3,0,1,1,1,0,1,3,0,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 6z^2+19z+15 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+10w^3z^2-4w^3z+23w^2z+15w |
| Inner characteristic polynomial | t^6+34t^4+69t^2 |
| Outer characteristic polynomial | t^7+50t^5+118t^3+7t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -1600*K1**2*K2**4 + 3360*K1**2*K2**3 - 6896*K1**2*K2**2 - 736*K1**2*K2*K4 + 4584*K1**2*K2 - 3112*K1**2 + 1984*K1*K2**3*K3 - 992*K1*K2**2*K3 - 224*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4752*K1*K2*K3 + 456*K1*K3*K4 + 8*K1*K4*K5 - 288*K2**6 + 160*K2**4*K4 - 2616*K2**4 - 816*K2**2*K3**2 - 8*K2**2*K4**2 + 1648*K2**2*K4 - 720*K2**2 + 328*K2*K3*K5 - 968*K3**2 - 326*K4**2 - 32*K5**2 + 2148 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]] |
| If K is slice | False |