| Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,0,1,1,3,4,0,1,1,2,1,2,3,-1,-1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.418'] |
| Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.315', '6.337', '6.389', '6.418', '6.599', '6.675', '6.686', '6.688', '6.746', '6.747', '6.809', '6.1034', '6.1128', '6.1133', '6.1334', '6.1363', '6.1489', '6.1539', '6.1564', '6.1821', '6.1863'] |
| Outer characteristic polynomial of the knot is: t^7+86t^5+153t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.418'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 224*K1**4*K2 - 832*K1**4 + 128*K1**3*K2**3*K3 + 224*K1**3*K2*K3 - 448*K1**2*K2**4 + 448*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 1952*K1**2*K2**2 + 2120*K1**2*K2 - 192*K1**2*K3**2 - 32*K1**2*K4**2 - 1176*K1**2 + 640*K1*K2**3*K3 + 32*K1*K2*K3**3 + 1640*K1*K2*K3 + 272*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 656*K2**4 - 416*K2**2*K3**2 - 16*K2**2*K4**2 + 216*K2**2*K4 - 526*K2**2 + 144*K2*K3*K5 + 8*K2*K4*K6 - 516*K3**2 - 140*K4**2 - 36*K5**2 - 2*K6**2 + 1114 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.418'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17081', 'vk6.17322', 'vk6.20052', 'vk6.20253', 'vk6.21184', 'vk6.21560', 'vk6.23465', 'vk6.26899', 'vk6.27117', 'vk6.27486', 'vk6.28654', 'vk6.29083', 'vk6.35598', 'vk6.38323', 'vk6.38510', 'vk6.38905', 'vk6.40464', 'vk6.41106', 'vk6.42975', 'vk6.45199', 'vk6.45410', 'vk6.45658', 'vk6.47024', 'vk6.47391', 'vk6.55220', 'vk6.56734', 'vk6.56866', 'vk6.57836', 'vk6.59620', 'vk6.61162', 'vk6.61395', 'vk6.61622', 'vk6.62404', 'vk6.62802', 'vk6.65023', 'vk6.66573', 'vk6.68295', 'vk6.69085', 'vk6.69225', 'vk6.69868'] |
| 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 | O1O2O3O4O5U6U1O6U2U4U5U3 |
| R3 orbit | {'O1O2O3O4O5U2U6U1O6U4U5U3', 'O1O2O3O4O5U6U1O6U2U4U5U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U3U1U2U4O6U5U6 |
| Gauss code of K* | O1O2O3O4U5U1U4U2U3O6O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6O5U2U3U1U4U6 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -3 -2 2 1 3 -1],[ 3 0 0 3 1 2 3],[ 2 0 0 3 1 2 2],[-2 -3 -3 0 -1 1 -2],[-1 -1 -1 1 0 1 -1],[-3 -2 -2 -1 -1 0 -3],[ 1 -3 -2 2 1 3 0]] |
| Primitive based matrix | [[ 0 3 2 1 -1 -2 -3],[-3 0 -1 -1 -3 -2 -2],[-2 1 0 -1 -2 -3 -3],[-1 1 1 0 -1 -1 -1],[ 1 3 2 1 0 -2 -3],[ 2 2 3 1 2 0 0],[ 3 2 3 1 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,1,2,3,1,1,3,2,2,1,2,3,3,1,1,1,2,3,0] |
| Phi over symmetry | [-3,-2,-1,1,2,3,0,1,1,3,4,0,1,1,2,1,2,3,-1,-1,1] |
| Phi of -K | [-3,-2,-1,1,2,3,1,-1,3,2,4,-1,2,1,3,1,1,1,0,1,0] |
| Phi of K* | [-3,-2,-1,1,2,3,0,1,1,3,4,0,1,1,2,1,2,3,-1,-1,1] |
| Phi of -K* | [-3,-2,-1,1,2,3,0,3,1,3,2,2,1,3,2,1,2,3,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 8z+17 |
| Enhanced Jones-Krushkal polynomial | -4w^3z+12w^2z+17w |
| Inner characteristic polynomial | t^6+58t^4+61t^2 |
| Outer characteristic polynomial | t^7+86t^5+153t^3 |
| Flat arrow polynomial | 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 224*K1**4*K2 - 832*K1**4 + 128*K1**3*K2**3*K3 + 224*K1**3*K2*K3 - 448*K1**2*K2**4 + 448*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 1952*K1**2*K2**2 + 2120*K1**2*K2 - 192*K1**2*K3**2 - 32*K1**2*K4**2 - 1176*K1**2 + 640*K1*K2**3*K3 + 32*K1*K2*K3**3 + 1640*K1*K2*K3 + 272*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 656*K2**4 - 416*K2**2*K3**2 - 16*K2**2*K4**2 + 216*K2**2*K4 - 526*K2**2 + 144*K2*K3*K5 + 8*K2*K4*K6 - 516*K3**2 - 140*K4**2 - 36*K5**2 - 2*K6**2 + 1114 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{4, 6}, {1, 5}, {2, 3}]] |
| If K is slice | True |