| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,1,2,4,1,0,1,1,-1,1,1,1,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1008'] |
| Arrow polynomial of the knot is: -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.235', '6.379', '6.411', '6.547', '6.811', '6.818', '6.823', '6.897', '6.898', '6.1008', '6.1053', '6.1109', '6.1110', '6.1130', '6.1222', '6.1239', '6.1303', '6.1307', '6.1342', '6.1491', '6.1495', '6.1496', '6.1519', '6.1592', '6.1593', '6.1642', '6.1652', '6.1653', '6.1671', '6.1673', '6.1717'] |
| Outer characteristic polynomial of the knot is: t^7+29t^5+54t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1008', '6.1128'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**6 + 1120*K1**4*K2 - 4624*K1**4 + 320*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1984*K1**3*K3 + 64*K1**2*K2**3 - 2704*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 10072*K1**2*K2 - 1008*K1**2*K3**2 - 128*K1**2*K3*K5 - 128*K1**2*K4**2 - 6392*K1**2 - 384*K1*K2**2*K3 - 32*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 7000*K1*K2*K3 + 1960*K1*K3*K4 + 232*K1*K4*K5 - 64*K2**4 - 144*K2**2*K3**2 - 16*K2**2*K4**2 + 664*K2**2*K4 - 5100*K2**2 + 256*K2*K3*K5 + 32*K2*K4*K6 - 2620*K3**2 - 836*K4**2 - 140*K5**2 - 12*K6**2 + 5330 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1008'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3670', 'vk6.3765', 'vk6.3956', 'vk6.4051', 'vk6.4483', 'vk6.4580', 'vk6.5869', 'vk6.5998', 'vk6.7153', 'vk6.7328', 'vk6.7419', 'vk6.7914', 'vk6.8035', 'vk6.9348', 'vk6.17915', 'vk6.18010', 'vk6.18759', 'vk6.24450', 'vk6.24884', 'vk6.25345', 'vk6.37498', 'vk6.43877', 'vk6.44231', 'vk6.44534', 'vk6.48294', 'vk6.48357', 'vk6.50075', 'vk6.50185', 'vk6.50559', 'vk6.50624', 'vk6.55858', 'vk6.60739'] |
| 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 | O1O2O3O4U2U4O5U3O6U5U1U6 |
| R3 orbit | {'O1O2O3O4U2U4O5U3O6U5U1U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U4U6O5U2O6U1U3 |
| Gauss code of K* | O1O2O3U2U4U5U6O4O6U1O5U3 |
| Gauss code of -K* | O1O2O3U1O4U3O5O6U5U4U6U2 |
| 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 -1 -2 0 1 0 2],[ 1 0 -2 0 1 1 2],[ 2 2 0 2 1 1 0],[ 0 0 -2 0 0 1 1],[-1 -1 -1 0 0 0 0],[ 0 -1 -1 -1 0 0 1],[-2 -2 0 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -1 -2 0],[-1 0 0 0 0 -1 -1],[ 0 1 0 0 1 0 -2],[ 0 1 0 -1 0 -1 -1],[ 1 2 1 0 1 0 -2],[ 2 0 1 2 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,1,1,2,0,0,0,1,1,-1,0,2,1,1,2] |
| Phi over symmetry | [-2,-1,0,0,1,2,-1,0,1,2,4,1,0,1,1,-1,1,1,1,1,1] |
| Phi of -K | [-2,-1,0,0,1,2,-1,0,1,2,4,1,0,1,1,-1,1,1,1,1,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,1,1,1,4,1,1,1,2,-1,0,1,1,0,-1] |
| Phi of -K* | [-2,-1,0,0,1,2,2,1,2,1,0,1,0,1,2,-1,0,1,0,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 20z+41 |
| Enhanced Jones-Krushkal polynomial | 20w^2z+41w |
| Inner characteristic polynomial | t^6+19t^4+20t^2 |
| Outer characteristic polynomial | t^7+29t^5+54t^3+6t |
| Flat arrow polynomial | -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5 |
| 2-strand cable arrow polynomial | -128*K1**6 + 1120*K1**4*K2 - 4624*K1**4 + 320*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1984*K1**3*K3 + 64*K1**2*K2**3 - 2704*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 10072*K1**2*K2 - 1008*K1**2*K3**2 - 128*K1**2*K3*K5 - 128*K1**2*K4**2 - 6392*K1**2 - 384*K1*K2**2*K3 - 32*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 7000*K1*K2*K3 + 1960*K1*K3*K4 + 232*K1*K4*K5 - 64*K2**4 - 144*K2**2*K3**2 - 16*K2**2*K4**2 + 664*K2**2*K4 - 5100*K2**2 + 256*K2*K3*K5 + 32*K2*K4*K6 - 2620*K3**2 - 836*K4**2 - 140*K5**2 - 12*K6**2 + 5330 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}]] |
| If K is slice | False |