| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,1,2,0,1,1,-1,1,1,0,1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1823'] |
| Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.241', '6.341', '6.542', '6.567', '6.699', '6.713', '6.771', '6.791', '6.1025', '6.1039', '6.1041', '6.1072', '6.1077', '6.1121', '6.1123', '6.1499', '6.1502', '6.1531', '6.1645', '6.1648', '6.1726', '6.1727', '6.1761', '6.1784', '6.1807', '6.1823', '6.1832', '6.1869', '6.1873', '6.1874'] |
| Outer characteristic polynomial of the knot is: t^7+25t^5+67t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1823'] |
| 2-strand cable arrow polynomial of the knot is: -448*K1**4*K2**2 + 928*K1**4*K2 - 2592*K1**4 + 480*K1**3*K2*K3 - 448*K1**3*K3 - 384*K1**2*K2**4 + 2208*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 8080*K1**2*K2**2 - 448*K1**2*K2*K4 + 8936*K1**2*K2 - 288*K1**2*K3**2 - 4512*K1**2 + 864*K1*K2**3*K3 - 1728*K1*K2**2*K3 - 96*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6952*K1*K2*K3 + 656*K1*K3*K4 + 40*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1848*K2**4 - 464*K2**2*K3**2 - 48*K2**2*K4**2 + 1720*K2**2*K4 - 3222*K2**2 + 256*K2*K3*K5 + 16*K2*K4*K6 - 1628*K3**2 - 438*K4**2 - 44*K5**2 - 2*K6**2 + 3788 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1823'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4680', 'vk6.4977', 'vk6.6152', 'vk6.6631', 'vk6.8157', 'vk6.8571', 'vk6.9543', 'vk6.9888', 'vk6.20692', 'vk6.22132', 'vk6.28213', 'vk6.29638', 'vk6.39669', 'vk6.41910', 'vk6.46253', 'vk6.47860', 'vk6.48720', 'vk6.48937', 'vk6.49502', 'vk6.49709', 'vk6.50746', 'vk6.50957', 'vk6.51229', 'vk6.51420', 'vk6.57627', 'vk6.58785', 'vk6.62307', 'vk6.63240', 'vk6.67097', 'vk6.67961', 'vk6.69697', 'vk6.70380'] |
| 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 | O1O2O3U1U2O4U5O6O5U3U6U4 |
| R3 orbit | {'O1O2O3U1U2O4U5O6O5U3U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5U1O6O5U6O4U2U3 |
| Gauss code of K* | O1O2O3U4U5U1O4O5U3O6U2U6 |
| Gauss code of -K* | O1O2O3U4U2O4U1O5O6U3U5U6 |
| 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 1 0],[ 2 0 1 2 1 2 1],[ 0 -1 0 1 1 0 1],[ 0 -2 -1 0 2 0 0],[-1 -1 -1 -2 0 0 -1],[-1 -2 0 0 0 0 0],[ 0 -1 -1 0 1 0 0]] |
| Primitive based matrix | [[ 0 1 1 0 0 0 -2],[-1 0 0 0 0 0 -2],[-1 0 0 -1 -1 -2 -1],[ 0 0 1 0 1 1 -1],[ 0 0 1 -1 0 0 -1],[ 0 0 2 -1 0 0 -2],[ 2 2 1 1 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,0,0,0,2,0,0,0,0,2,1,1,2,1,-1,-1,1,0,1,2] |
| Phi over symmetry | [-2,0,0,0,1,1,0,1,1,1,2,0,1,1,-1,1,1,0,1,0,0] |
| Phi of -K | [-2,0,0,0,1,1,0,1,1,1,2,0,1,1,-1,1,1,0,1,0,0] |
| Phi of K* | [-1,-1,0,0,0,2,0,-1,0,0,2,1,1,1,1,-1,0,0,1,1,1] |
| Phi of -K* | [-2,0,0,0,1,1,1,1,2,1,2,-1,0,1,0,1,1,0,2,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 4z^2+23z+31 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+23w^2z+31w |
| Inner characteristic polynomial | t^6+19t^4+36t^2+4 |
| Outer characteristic polynomial | t^7+25t^5+67t^3+9t |
| Flat arrow polynomial | 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| 2-strand cable arrow polynomial | -448*K1**4*K2**2 + 928*K1**4*K2 - 2592*K1**4 + 480*K1**3*K2*K3 - 448*K1**3*K3 - 384*K1**2*K2**4 + 2208*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 8080*K1**2*K2**2 - 448*K1**2*K2*K4 + 8936*K1**2*K2 - 288*K1**2*K3**2 - 4512*K1**2 + 864*K1*K2**3*K3 - 1728*K1*K2**2*K3 - 96*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6952*K1*K2*K3 + 656*K1*K3*K4 + 40*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1848*K2**4 - 464*K2**2*K3**2 - 48*K2**2*K4**2 + 1720*K2**2*K4 - 3222*K2**2 + 256*K2*K3*K5 + 16*K2*K4*K6 - 1628*K3**2 - 438*K4**2 - 44*K5**2 - 2*K6**2 + 3788 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | False |