| Min(phi) over symmetries of the knot is: [-3,-2,1,1,3,0,1,3,4,0,1,2,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['5.32'] |
| Arrow polynomial of the knot is: -8*K1**3 + 2*K1**2 + 4*K1*K2 + 4*K1 - K2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['5.32', '5.83', '7.4858', '7.4981', '7.6735', '7.7680', '7.7682', '7.7771', '7.7847', '7.8811', '7.10033', '7.11951', '7.13003', '7.13963', '7.13973', '7.16900', '7.16978', '7.17546', '7.17553', '7.17796', '7.17807', '7.20662', '7.21135', '7.21138', '7.21166', '7.21191', '7.21197', '7.21594', '7.21788', '7.22143', '7.22227', '7.22238', '7.22248', '7.22632', '7.22819', '7.23134', '7.24160', '7.24922', '7.25324', '7.25443', '7.25461', '7.25595', '7.25791', '7.26158', '7.26697', '7.27195', '7.27272', '7.29047', '7.30002', '7.30072', '7.30429', '7.30504', '7.30686', '7.30916', '7.31802', '7.32058', '7.32073', '7.32181', '7.33605', '7.33653', '7.34366', '7.34399', '7.34980', '7.35049', '7.35123', '7.35155', '7.35184', '7.36840', '7.37252', '7.37409', '7.37881', '7.37910', '7.37967', '7.38038', '7.38335', '7.38352', '7.38377', '7.38452', '7.38507', '7.38543', '7.38737', '7.38837', '7.39289', '7.39339', '7.39518', '7.40318', '7.41784', '7.42205', '7.42530', '7.42551', '7.42809', '7.42827', '7.43071', '7.43089', '7.43314', '7.43528'] |
| Outer characteristic polynomial of the knot is: t^6+56t^4+43t^2+1 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.32', '7.3755'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**2*K2**4 + 512*K1**2*K2**3 - 1488*K1**2*K2**2 - 64*K1**2*K2*K4 + 1360*K1**2*K2 - 936*K1**2 + 288*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 352*K1*K2**2*K3 - 32*K1*K2**2*K5 + 1168*K1*K2*K3 + 48*K1*K3*K4 - 64*K2**6 + 64*K2**4*K4 - 632*K2**4 - 144*K2**2*K3**2 - 48*K2**2*K4**2 + 520*K2**2*K4 - 408*K2**2 + 32*K2*K3*K5 + 8*K2*K4*K6 - 232*K3**2 - 86*K4**2 + 644 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.32'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.431', 'vk5.459', 'vk5.636', 'vk5.778', 'vk5.924', 'vk5.962', 'vk5.1144', 'vk5.1297', 'vk5.1487', 'vk5.1526', 'vk5.1647', 'vk5.1765', 'vk5.1796', 'vk5.1811', 'vk5.1865', 'vk5.1931'] |
| 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 | O1O2O3O4U2O5U1U5U3U4 |
| R3 orbit | {'O1O2O3O4U2O5U1U5U3U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U2U5U4O5U3 |
| Gauss code of K* | O1O2O3O4U1U5U3U4O5U2 |
| Gauss code of -K* | O1O2O3O4U3O5U1U2U5U4 |
| 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 3 1],[ 3 0 0 3 4 1],[ 2 0 0 1 2 0],[-1 -3 -1 0 1 0],[-3 -4 -2 -1 0 0],[-1 -1 0 0 0 0]] |
| Primitive based matrix | [[ 0 3 1 1 -2 -3],[-3 0 0 -1 -2 -4],[-1 0 0 0 0 -1],[-1 1 0 0 -1 -3],[ 2 2 0 1 0 0],[ 3 4 1 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,2,3,0,1,2,4,0,0,1,1,3,0] |
| Phi over symmetry | [-3,-2,1,1,3,0,1,3,4,0,1,2,0,0,1] |
| Phi of -K | [-3,-2,1,1,3,1,1,3,2,2,3,3,0,1,2] |
| Phi of K* | [-3,-1,-1,2,3,1,2,3,2,0,2,1,3,3,1] |
| Phi of -K* | [-3,-2,1,1,3,0,1,3,4,0,1,2,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | -4z^2-15z-13 |
| Enhanced Jones-Krushkal polynomial | -4w^3z^2-15w^2z-13w |
| Inner characteristic polynomial | t^5+32t^3+10t |
| Outer characteristic polynomial | t^6+56t^4+43t^2+1 |
| Flat arrow polynomial | -8*K1**3 + 2*K1**2 + 4*K1*K2 + 4*K1 - K2 |
| 2-strand cable arrow polynomial | -192*K1**2*K2**4 + 512*K1**2*K2**3 - 1488*K1**2*K2**2 - 64*K1**2*K2*K4 + 1360*K1**2*K2 - 936*K1**2 + 288*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 352*K1*K2**2*K3 - 32*K1*K2**2*K5 + 1168*K1*K2*K3 + 48*K1*K3*K4 - 64*K2**6 + 64*K2**4*K4 - 632*K2**4 - 144*K2**2*K3**2 - 48*K2**2*K4**2 + 520*K2**2*K4 - 408*K2**2 + 32*K2*K3*K5 + 8*K2*K4*K6 - 232*K3**2 - 86*K4**2 + 644 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 5}, {2, 4}, {3}], [{1, 5}, {3, 4}, {2}], [{1, 5}, {4}, {2, 3}], [{1, 5}, {4}, {3}, {2}], [{2, 5}, {1, 4}, {3}], [{2, 5}, {3, 4}, {1}], [{2, 5}, {4}, {1, 3}], [{2, 5}, {4}, {3}, {1}], [{3, 5}, {1, 4}, {2}], [{3, 5}, {2, 4}, {1}], [{3, 5}, {4}, {1, 2}], [{3, 5}, {4}, {2}, {1}], [{4, 5}, {1, 3}, {2}], [{4, 5}, {2, 3}, {1}], [{4, 5}, {3}, {1, 2}], [{4, 5}, {3}, {2}, {1}], [{5}, {1, 4}, {2, 3}], [{5}, {1, 4}, {3}, {2}], [{5}, {2, 4}, {1, 3}], [{5}, {2, 4}, {3}, {1}], [{5}, {3, 4}, {1, 2}], [{5}, {3, 4}, {2}, {1}], [{5}, {4}, {1, 3}, {2}], [{5}, {4}, {2, 3}, {1}], [{5}, {4}, {3}, {1, 2}]] |
| If K is slice | False |