| Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,-1,2,1,4,4,1,0,2,1,0,1,2,-1,0,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.771'] |
| Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.241', '6.341', '6.542', '6.567', '6.699', '6.713', '6.771', '6.791', '6.1025', '6.1039', '6.1041', '6.1072', '6.1077', '6.1121', '6.1123', '6.1499', '6.1502', '6.1531', '6.1645', '6.1648', '6.1726', '6.1727', '6.1761', '6.1784', '6.1807', '6.1823', '6.1832', '6.1869', '6.1873', '6.1874'] |
| Outer characteristic polynomial of the knot is: t^7+78t^5+61t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.771'] |
| 2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 960*K1**4*K2 - 2096*K1**4 + 448*K1**3*K2*K3 + 32*K1**3*K3*K4 - 800*K1**3*K3 - 320*K1**2*K2**4 + 992*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 4016*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 224*K1**2*K2*K4 + 6760*K1**2*K2 - 560*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 4272*K1**2 + 480*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 832*K1*K2**2*K3 - 96*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4472*K1*K2*K3 + 712*K1*K3*K4 + 40*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 584*K2**4 - 272*K2**2*K3**2 - 16*K2**2*K4**2 + 568*K2**2*K4 - 2838*K2**2 + 128*K2*K3*K5 + 8*K2*K4*K6 - 1224*K3**2 - 278*K4**2 - 24*K5**2 - 2*K6**2 + 3140 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.771'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4435', 'vk6.4532', 'vk6.5821', 'vk6.5950', 'vk6.7876', 'vk6.7987', 'vk6.9300', 'vk6.9421', 'vk6.10174', 'vk6.10247', 'vk6.10388', 'vk6.17875', 'vk6.17938', 'vk6.18270', 'vk6.18605', 'vk6.24378', 'vk6.25156', 'vk6.30057', 'vk6.30120', 'vk6.36888', 'vk6.37346', 'vk6.43805', 'vk6.44105', 'vk6.44428', 'vk6.48625', 'vk6.50523', 'vk6.50608', 'vk6.51130', 'vk6.51675', 'vk6.55842', 'vk6.56073', 'vk6.65504'] |
| 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 | O1O2O3O4U3O5U1U6U5O6U2U4 |
| R3 orbit | {'O1O2O3O4U3O5U1U6U5O6U2U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U3O5U6U5U4O6U2 |
| Gauss code of K* | O1O2O3U2O4O5U1U4U6U5O6U3 |
| Gauss code of -K* | O1O2O3U1O4U5U4U6U3O5O6U2 |
| 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 0 -1 3 2 -1],[ 3 0 2 0 4 1 2],[ 0 -2 0 0 2 1 -1],[ 1 0 0 0 1 0 1],[-3 -4 -2 -1 0 1 -4],[-2 -1 -1 0 -1 0 -2],[ 1 -2 1 -1 4 2 0]] |
| Primitive based matrix | [[ 0 3 2 0 -1 -1 -3],[-3 0 1 -2 -1 -4 -4],[-2 -1 0 -1 0 -2 -1],[ 0 2 1 0 0 -1 -2],[ 1 1 0 0 0 1 0],[ 1 4 2 1 -1 0 -2],[ 3 4 1 2 0 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,1,1,3,-1,2,1,4,4,1,0,2,1,0,1,2,-1,0,2] |
| Phi over symmetry | [-3,-2,0,1,1,3,-1,2,1,4,4,1,0,2,1,0,1,2,-1,0,2] |
| Phi of -K | [-3,-1,-1,0,2,3,0,2,1,4,2,1,0,1,0,1,3,3,1,1,2] |
| Phi of K* | [-3,-2,0,1,1,3,2,1,0,3,2,1,1,3,4,0,1,1,-1,0,2] |
| Phi of -K* | [-3,-1,-1,0,2,3,0,2,2,1,4,1,0,0,1,1,2,4,1,2,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 2z^2+19z+31 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+19w^2z+31w |
| Inner characteristic polynomial | t^6+54t^4+32t^2+1 |
| Outer characteristic polynomial | t^7+78t^5+61t^3+5t |
| Flat arrow polynomial | 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| 2-strand cable arrow polynomial | -320*K1**4*K2**2 + 960*K1**4*K2 - 2096*K1**4 + 448*K1**3*K2*K3 + 32*K1**3*K3*K4 - 800*K1**3*K3 - 320*K1**2*K2**4 + 992*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 4016*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 224*K1**2*K2*K4 + 6760*K1**2*K2 - 560*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 4272*K1**2 + 480*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 832*K1*K2**2*K3 - 96*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4472*K1*K2*K3 + 712*K1*K3*K4 + 40*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 584*K2**4 - 272*K2**2*K3**2 - 16*K2**2*K4**2 + 568*K2**2*K4 - 2838*K2**2 + 128*K2*K3*K5 + 8*K2*K4*K6 - 1224*K3**2 - 278*K4**2 - 24*K5**2 - 2*K6**2 + 3140 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{6}, {1, 5}, {3, 4}, {2}]] |
| If K is slice | False |