| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,1,2,3,0,1,1,1,0,1,0,0,-1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1072'] |
| 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+22t^5+44t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1072'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**6 + 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 2368*K1**4*K2 - 3600*K1**4 - 256*K1**3*K2**2*K3 + 736*K1**3*K2*K3 - 928*K1**3*K3 - 448*K1**2*K2**4 + 2976*K1**2*K2**3 - 8512*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 9504*K1**2*K2 - 336*K1**2*K3**2 - 64*K1**2*K3*K5 - 32*K1**2*K4**2 - 4996*K1**2 + 768*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 32*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 6784*K1*K2*K3 + 952*K1*K3*K4 + 208*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1976*K2**4 - 400*K2**2*K3**2 - 48*K2**2*K4**2 + 1488*K2**2*K4 - 3174*K2**2 + 304*K2*K3*K5 + 16*K2*K4*K6 - 1700*K3**2 - 590*K4**2 - 160*K5**2 - 2*K6**2 + 4252 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1072'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11049', 'vk6.11127', 'vk6.12211', 'vk6.12318', 'vk6.16418', 'vk6.19240', 'vk6.19330', 'vk6.19533', 'vk6.19625', 'vk6.22723', 'vk6.22822', 'vk6.26050', 'vk6.26094', 'vk6.26429', 'vk6.26518', 'vk6.30618', 'vk6.30713', 'vk6.31920', 'vk6.34765', 'vk6.38114', 'vk6.38128', 'vk6.42382', 'vk6.44639', 'vk6.44748', 'vk6.51854', 'vk6.52715', 'vk6.52818', 'vk6.56583', 'vk6.56626', 'vk6.64711', 'vk6.66279', 'vk6.66292'] |
| 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 | O1O2O3O4U5U4O6U1O5U3U6U2 |
| R3 orbit | {'O1O2O3O4U5U4O6U1O5U3U6U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U3U5U2O6U4O5U1U6 |
| Gauss code of K* | O1O2O3U4U3U1U5O6O5U2O4U6 |
| Gauss code of -K* | O1O2O3U4O5U2O6O4U6U3U1U5 |
| 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 -1 1],[ 2 0 2 0 0 1 1],[-1 -2 0 -1 1 -1 0],[ 0 0 1 0 1 -1 0],[-1 0 -1 -1 0 -1 -1],[ 1 -1 1 1 1 0 1],[-1 -1 0 0 1 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -1],[-1 -1 0 -1 -1 -1 0],[-1 0 1 0 -1 -1 -2],[ 0 0 1 1 0 -1 0],[ 1 1 1 1 1 0 -1],[ 2 1 0 2 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,0,1,1,1,1,1,0,1,1,2,1,0,1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,2,1,2,3,0,1,1,1,0,1,0,0,-1,-1] |
| Phi of -K | [-2,-1,0,1,1,1,0,2,1,2,3,0,1,1,1,0,1,0,0,-1,-1] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,-1,0,1,3,0,0,1,1,1,1,2,0,2,0] |
| Phi of -K* | [-2,-1,0,1,1,1,1,0,0,1,2,1,1,1,1,1,0,1,-1,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 2z^2+19z+31 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2-4w^3z+23w^2z+31w |
| Inner characteristic polynomial | t^6+14t^4+23t^2 |
| Outer characteristic polynomial | t^7+22t^5+44t^3+8t |
| Flat arrow polynomial | 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| 2-strand cable arrow polynomial | -256*K1**6 + 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 2368*K1**4*K2 - 3600*K1**4 - 256*K1**3*K2**2*K3 + 736*K1**3*K2*K3 - 928*K1**3*K3 - 448*K1**2*K2**4 + 2976*K1**2*K2**3 - 8512*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 9504*K1**2*K2 - 336*K1**2*K3**2 - 64*K1**2*K3*K5 - 32*K1**2*K4**2 - 4996*K1**2 + 768*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 32*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 6784*K1*K2*K3 + 952*K1*K3*K4 + 208*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1976*K2**4 - 400*K2**2*K3**2 - 48*K2**2*K4**2 + 1488*K2**2*K4 - 3174*K2**2 + 304*K2*K3*K5 + 16*K2*K4*K6 - 1700*K3**2 - 590*K4**2 - 160*K5**2 - 2*K6**2 + 4252 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}]] |
| If K is slice | False |