| Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,2,2,4,0,1,2,2,-1,1,-1,1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.630'] |
| Arrow polynomial of the knot is: -2*K1*K2 + K1 + K3 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.1', '4.3', '6.59', '6.66', '6.112', '6.215', '6.297', '6.306', '6.346', '6.351', '6.352', '6.353', '6.368', '6.393', '6.398', '6.402', '6.420', '6.422', '6.524', '6.529', '6.630', '6.632', '6.633', '6.642', '6.684', '6.707', '6.708', '6.717', '6.719', '6.721', '6.722', '6.737', '6.793', '6.835', '6.837', '6.847', '6.849', '6.857', '6.858', '6.883', '6.902', '6.913', '6.1084', '6.1092', '6.1097', '6.1136', '6.1146', '6.1155', '6.1159', '6.1374', '7.349', '7.365', '7.690', '7.2260', '7.2269', '7.2612', '7.2624', '7.2972', '7.2975', '7.4214', '7.4542', '7.4546', '7.9686', '7.9695', '7.9947', '7.10639', '7.10643', '7.10829', '7.10833', '7.13433', '7.15124', '7.15128', '7.15638', '7.15647', '7.15703', '7.15845', '7.16115', '7.16120', '7.16150', '7.19418', '7.19470', '7.19474', '7.19871', '7.20310', '7.20362', '7.20421', '7.20424', '7.23942', '7.24011', '7.24100', '7.24114', '7.24116', '7.24445', '7.26258', '7.26318', '7.26811', '7.26827', '7.27967', '7.28040', '7.28124', '7.28138', '7.29092', '7.29107', '7.29452', '7.29853', '7.30091', '7.30098', '7.30140', '7.30193', '7.30339', '7.30350', '7.30354'] |
| Outer characteristic polynomial of the knot is: t^7+66t^5+109t^3+16t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.630'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4 + 224*K1**3*K2*K3 - 64*K1**3*K3 - 2368*K1**2*K2**2 - 192*K1**2*K2*K4 + 2096*K1**2*K2 - 416*K1**2*K3**2 - 2616*K1**2 + 192*K1*K2**3*K3 - 32*K1*K2**2*K3 - 128*K1*K2**2*K5 + 4792*K1*K2*K3 + 528*K1*K3*K4 + 40*K1*K4*K5 + 24*K1*K5*K6 - 704*K2**4 - 384*K2**2*K3**2 - 8*K2**2*K4**2 + 512*K2**2*K4 - 1966*K2**2 + 480*K2*K3*K5 + 16*K2*K4*K6 - 1880*K3**2 - 228*K4**2 - 168*K5**2 - 18*K6**2 + 2394 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.630'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73748', 'vk6.73759', 'vk6.73889', 'vk6.73898', 'vk6.75693', 'vk6.75707', 'vk6.75891', 'vk6.75900', 'vk6.78683', 'vk6.78695', 'vk6.78882', 'vk6.78893', 'vk6.80308', 'vk6.80319', 'vk6.80434', 'vk6.80443', 'vk6.81699', 'vk6.81711', 'vk6.81712', 'vk6.81800', 'vk6.82198', 'vk6.82454', 'vk6.82460', 'vk6.82473', 'vk6.82474', 'vk6.84428', 'vk6.84447', 'vk6.84450', 'vk6.87779', 'vk6.88109', 'vk6.88398', 'vk6.89641'] |
| 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 | O1O2O3O4U1O5U2U4O6U5U3U6 |
| R3 orbit | {'O1O2O3O4U1O5U2U4O6U5U3U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U2U6O5U1U3O6U4 |
| Gauss code of K* | O1O2O3U4U5U2U6O4U1O5O6U3 |
| Gauss code of -K* | O1O2O3U1O4O5U3O6U4U2U5U6 |
| 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 -2 1 1 1 2],[ 3 0 1 3 2 2 1],[ 2 -1 0 3 1 2 2],[-1 -3 -3 0 -1 1 2],[-1 -2 -1 1 0 1 1],[-1 -2 -2 -1 -1 0 1],[-2 -1 -2 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 1 -2 -3],[-2 0 -1 -1 -2 -2 -1],[-1 1 0 1 1 -1 -2],[-1 1 -1 0 -1 -2 -2],[-1 2 -1 1 0 -3 -3],[ 2 2 1 2 3 0 -1],[ 3 1 2 2 3 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,-1,2,3,1,1,2,2,1,-1,-1,1,2,1,2,2,3,3,1] |
| Phi over symmetry | [-3,-2,1,1,1,2,0,1,2,2,4,0,1,2,2,-1,1,-1,1,0,0] |
| Phi of -K | [-3,-2,1,1,1,2,0,1,2,2,4,0,1,2,2,-1,1,-1,1,0,0] |
| Phi of K* | [-2,-1,-1,-1,2,3,-1,0,0,2,4,-1,1,0,1,1,2,2,1,2,0] |
| Phi of -K* | [-3,-2,1,1,1,2,1,2,2,3,1,1,2,3,2,1,1,1,-1,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-3t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+5w^3z^2-8w^3z+24w^2z+21w |
| Inner characteristic polynomial | t^6+46t^4+21t^2+1 |
| Outer characteristic polynomial | t^7+66t^5+109t^3+16t |
| Flat arrow polynomial | -2*K1*K2 + K1 + K3 + 1 |
| 2-strand cable arrow polynomial | -128*K1**4 + 224*K1**3*K2*K3 - 64*K1**3*K3 - 2368*K1**2*K2**2 - 192*K1**2*K2*K4 + 2096*K1**2*K2 - 416*K1**2*K3**2 - 2616*K1**2 + 192*K1*K2**3*K3 - 32*K1*K2**2*K3 - 128*K1*K2**2*K5 + 4792*K1*K2*K3 + 528*K1*K3*K4 + 40*K1*K4*K5 + 24*K1*K5*K6 - 704*K2**4 - 384*K2**2*K3**2 - 8*K2**2*K4**2 + 512*K2**2*K4 - 1966*K2**2 + 480*K2*K3*K5 + 16*K2*K4*K6 - 1880*K3**2 - 228*K4**2 - 168*K5**2 - 18*K6**2 + 2394 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]] |
| If K is slice | False |