| Min(phi) over symmetries of the knot is: [-3,-3,0,2,2,2,-1,1,2,3,4,1,1,2,3,1,1,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.213'] |
| Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.120', '6.213', '6.216', '6.320', '6.322', '6.615', '6.617', '6.891', '6.951', '6.955', '6.1001', '6.1012', '6.1022', '6.1043', '6.1047', '6.1063', '6.1074', '6.1249', '6.1544', '6.1546', '6.1555', '6.1573', '6.1574', '6.1585', '6.1756', '6.1757', '6.1762', '6.1802', '6.1803', '6.1824', '6.1881', '6.1935'] |
| Outer characteristic polynomial of the knot is: t^7+81t^5+29t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.213'] |
| 2-strand cable arrow polynomial of the knot is: -464*K1**2*K2**2 + 264*K1**2*K2 - 16*K1**2*K3**2 - 476*K1**2 + 160*K1*K2**3*K3 + 944*K1*K2*K3 + 80*K1*K3*K4 + 48*K1*K4*K5 - 104*K2**4 - 192*K2**2*K3**2 - 16*K2**2*K4**2 + 48*K2**2*K4 - 344*K2**2 + 104*K2*K3*K5 + 16*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 404*K3**2 - 78*K4**2 - 48*K5**2 - 8*K6**2 + 484 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.213'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70496', 'vk6.70509', 'vk6.70553', 'vk6.70574', 'vk6.70698', 'vk6.70725', 'vk6.70805', 'vk6.70826', 'vk6.70977', 'vk6.70992', 'vk6.71057', 'vk6.71078', 'vk6.71196', 'vk6.71211', 'vk6.71273', 'vk6.71284', 'vk6.71466', 'vk6.71518', 'vk6.71530', 'vk6.71768', 'vk6.71989', 'vk6.72007', 'vk6.72044', 'vk6.72060', 'vk6.72187', 'vk6.73660', 'vk6.75753', 'vk6.75794', 'vk6.75801', 'vk6.77088', 'vk6.77425', 'vk6.77441', 'vk6.77557', 'vk6.78110', 'vk6.78782', 'vk6.78789', 'vk6.79071', 'vk6.80379', 'vk6.81265', 'vk6.81285', 'vk6.86880', 'vk6.87014', 'vk6.87731', 'vk6.87799', 'vk6.87945', 'vk6.89135', 'vk6.89152', 'vk6.89508'] |
| 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 | O1O2O3O4O5U2O6U5U1U6U4U3 |
| R3 orbit | {'O1O2O3O4O5U2O6U5U1U6U4U3', 'O1O2O3O4O5U2U4O6U1U5U6U3', 'O1O2O3O4U1O5O6U2U5U6U4U3'} |
| R3 orbit length | 3 |
| Gauss code of -K | O1O2O3O4O5U3U2U6U5U1O6U4 |
| Gauss code of K* | O1O2O3O4O5U2U6U5U4U1O6U3 |
| Gauss code of -K* | O1O2O3O4O5U3O6U5U2U1U6U4 |
| 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 -3 2 2 0 2],[ 3 0 -1 4 3 1 2],[ 3 1 0 3 2 1 1],[-2 -4 -3 0 0 -1 1],[-2 -3 -2 0 0 -1 1],[ 0 -1 -1 1 1 0 1],[-2 -2 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 2 2 0 -3 -3],[-2 0 1 0 -1 -2 -3],[-2 -1 0 -1 -1 -1 -2],[-2 0 1 0 -1 -3 -4],[ 0 1 1 1 0 -1 -1],[ 3 2 1 3 1 0 1],[ 3 3 2 4 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-2,0,3,3,-1,0,1,2,3,1,1,1,2,1,3,4,1,1,-1] |
| Phi over symmetry | [-3,-3,0,2,2,2,-1,1,2,3,4,1,1,2,3,1,1,1,-1,-1,0] |
| Phi of -K | [-3,-3,0,2,2,2,-1,2,2,3,4,2,1,2,3,1,1,1,0,-1,-1] |
| Phi of K* | [-2,-2,-2,0,3,3,-1,-1,1,3,4,0,1,1,2,1,2,3,2,2,-1] |
| Phi of -K* | [-3,-3,0,2,2,2,-1,1,2,3,4,1,1,2,3,1,1,1,-1,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 2t^3-3t^2 |
| Normalized Jones-Krushkal polynomial | 3z+7 |
| Enhanced Jones-Krushkal polynomial | -6w^3z+9w^2z+7w |
| Inner characteristic polynomial | t^6+51t^4+8t^2 |
| Outer characteristic polynomial | t^7+81t^5+29t^3 |
| Flat arrow polynomial | -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| 2-strand cable arrow polynomial | -464*K1**2*K2**2 + 264*K1**2*K2 - 16*K1**2*K3**2 - 476*K1**2 + 160*K1*K2**3*K3 + 944*K1*K2*K3 + 80*K1*K3*K4 + 48*K1*K4*K5 - 104*K2**4 - 192*K2**2*K3**2 - 16*K2**2*K4**2 + 48*K2**2*K4 - 344*K2**2 + 104*K2*K3*K5 + 16*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 404*K3**2 - 78*K4**2 - 48*K5**2 - 8*K6**2 + 484 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {4, 5}, {1, 3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {3, 5}, {2, 4}, {1}]] |
| If K is slice | False |