| Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,2,2,3,0,1,1,1,1,1,1,1,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.794'] |
| Arrow polynomial of the knot is: -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.136', '6.207', '6.342', '6.370', '6.376', '6.442', '6.456', '6.539', '6.631', '6.636', '6.674', '6.679', '6.705', '6.740', '6.760', '6.794', '6.795', '6.1369'] |
| Outer characteristic polynomial of the knot is: t^7+46t^5+24t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.794'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**6 - 384*K1**4*K2**2 + 704*K1**4*K2 - 1472*K1**4 + 256*K1**3*K2*K3 - 384*K1**3*K3 + 448*K1**2*K2**3 - 2384*K1**2*K2**2 - 416*K1**2*K2*K4 + 3320*K1**2*K2 - 32*K1**2*K3**2 - 1392*K1**2 + 160*K1*K2**3*K3 - 96*K1*K2**2*K3 - 128*K1*K2**2*K5 + 2168*K1*K2*K3 + 192*K1*K3*K4 + 8*K1*K4*K5 - 384*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 392*K2**2*K4 - 1126*K2**2 + 96*K2*K3*K5 + 8*K2*K4*K6 - 436*K3**2 - 116*K4**2 - 20*K5**2 - 2*K6**2 + 1234 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.794'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17010', 'vk6.17252', 'vk6.19985', 'vk6.20242', 'vk6.21157', 'vk6.21546', 'vk6.23418', 'vk6.26871', 'vk6.27008', 'vk6.27461', 'vk6.28635', 'vk6.29059', 'vk6.35488', 'vk6.38306', 'vk6.38414', 'vk6.38876', 'vk6.40433', 'vk6.41076', 'vk6.42917', 'vk6.45170', 'vk6.45300', 'vk6.45641', 'vk6.47008', 'vk6.47378', 'vk6.55196', 'vk6.56720', 'vk6.56799', 'vk6.57811', 'vk6.58212', 'vk6.59579', 'vk6.61303', 'vk6.62383', 'vk6.64991', 'vk6.66416', 'vk6.66507', 'vk6.67182', 'vk6.67550', 'vk6.68275', 'vk6.69153', 'vk6.69852'] |
| 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 | O1O2O3O4U5O6U2U6U1O5U3U4 |
| R3 orbit | {'O1O2O3O4U5O6U2U6U1O5U3U4', 'O1O2O3O4U5U1O6U2U6O5U3U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U1U2O5U4U6U3O6U5 |
| Gauss code of K* | O1O2O3U4O5O6U3U1U5U6O4U2 |
| Gauss code of -K* | O1O2O3U2O4U5U6U3U1O5O6U4 |
| 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 1 3 -2 1],[ 1 0 -1 1 2 0 1],[ 2 1 0 1 2 1 1],[-1 -1 -1 0 1 -1 0],[-3 -2 -2 -1 0 -3 0],[ 2 0 -1 1 3 0 1],[-1 -1 -1 0 0 -1 0]] |
| Primitive based matrix | [[ 0 3 1 1 -1 -2 -2],[-3 0 0 -1 -2 -2 -3],[-1 0 0 0 -1 -1 -1],[-1 1 0 0 -1 -1 -1],[ 1 2 1 1 0 -1 0],[ 2 2 1 1 1 0 1],[ 2 3 1 1 0 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,1,2,2,0,1,2,2,3,0,1,1,1,1,1,1,1,0,-1] |
| Phi over symmetry | [-3,-1,-1,1,2,2,0,1,2,2,3,0,1,1,1,1,1,1,1,0,-1] |
| Phi of -K | [-2,-2,-1,1,1,3,-1,0,2,2,3,1,2,2,2,1,1,2,0,1,2] |
| Phi of K* | [-3,-1,-1,1,2,2,1,2,2,2,3,0,1,2,2,1,2,2,1,0,-1] |
| Phi of -K* | [-2,-2,-1,1,1,3,-1,0,1,1,3,1,1,1,2,1,1,2,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+2t^2-t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+26t^4+8t^2 |
| Outer characteristic polynomial | t^7+46t^5+24t^3+3t |
| Flat arrow polynomial | -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -128*K1**6 - 384*K1**4*K2**2 + 704*K1**4*K2 - 1472*K1**4 + 256*K1**3*K2*K3 - 384*K1**3*K3 + 448*K1**2*K2**3 - 2384*K1**2*K2**2 - 416*K1**2*K2*K4 + 3320*K1**2*K2 - 32*K1**2*K3**2 - 1392*K1**2 + 160*K1*K2**3*K3 - 96*K1*K2**2*K3 - 128*K1*K2**2*K5 + 2168*K1*K2*K3 + 192*K1*K3*K4 + 8*K1*K4*K5 - 384*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 392*K2**2*K4 - 1126*K2**2 + 96*K2*K3*K5 + 8*K2*K4*K6 - 436*K3**2 - 116*K4**2 - 20*K5**2 - 2*K6**2 + 1234 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | False |