| Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,0,2,1,2,4,1,1,1,2,1,0,2,-1,-1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.341'] |
| 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+64t^5+59t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.341'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 288*K1**4*K2 - 1120*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 384*K1**3*K2*K3 - 256*K1**3*K3 - 448*K1**2*K2**4 + 2048*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 6320*K1**2*K2**2 + 160*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 7720*K1**2*K2 - 480*K1**2*K3**2 - 32*K1**2*K4**2 - 5912*K1**2 + 864*K1*K2**3*K3 - 1184*K1*K2**2*K3 + 32*K1*K2*K3**3 - 128*K1*K2*K3*K4 + 6536*K1*K2*K3 + 1152*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1400*K2**4 - 624*K2**2*K3**2 - 16*K2**2*K4**2 + 1104*K2**2*K4 - 3374*K2**2 + 240*K2*K3*K5 + 8*K2*K4*K6 - 1920*K3**2 - 542*K4**2 - 56*K5**2 - 2*K6**2 + 4220 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.341'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11222', 'vk6.11301', 'vk6.12483', 'vk6.12594', 'vk6.18232', 'vk6.18569', 'vk6.24703', 'vk6.25118', 'vk6.30896', 'vk6.31019', 'vk6.32080', 'vk6.32199', 'vk6.36820', 'vk6.37283', 'vk6.44063', 'vk6.44404', 'vk6.51980', 'vk6.52075', 'vk6.52861', 'vk6.52908', 'vk6.56025', 'vk6.56301', 'vk6.60573', 'vk6.60913', 'vk6.63640', 'vk6.63685', 'vk6.64068', 'vk6.64113', 'vk6.65684', 'vk6.65976', 'vk6.68732', 'vk6.68942'] |
| 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 | O1O2O3O4O5U3U1O6U5U6U2U4 |
| R3 orbit | {'O1O2O3O4O5U3U1O6U5U6U2U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U2U4U6U1O6U5U3 |
| Gauss code of K* | O1O2O3O4U5U3U6U4U1O6O5U2 |
| Gauss code of -K* | O1O2O3O4U3O5O6U4U1U6U2U5 |
| 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 -2 3 1 1],[ 3 0 2 0 4 2 1],[ 0 -2 0 -1 2 0 1],[ 2 0 1 0 2 1 1],[-3 -4 -2 -2 0 -1 1],[-1 -2 0 -1 1 0 1],[-1 -1 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 1 0 -2 -3],[-3 0 1 -1 -2 -2 -4],[-1 -1 0 -1 -1 -1 -1],[-1 1 1 0 0 -1 -2],[ 0 2 1 0 0 -1 -2],[ 2 2 1 1 1 0 0],[ 3 4 1 2 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,0,2,3,-1,1,2,2,4,1,1,1,1,0,1,2,1,2,0] |
| Phi over symmetry | [-3,-2,0,1,1,3,0,2,1,2,4,1,1,1,2,1,0,2,-1,-1,1] |
| Phi of -K | [-3,-2,0,1,1,3,1,1,2,3,2,1,2,2,3,1,0,1,-1,1,3] |
| Phi of K* | [-3,-1,-1,0,2,3,1,3,1,3,2,1,1,2,2,0,2,3,1,1,1] |
| Phi of -K* | [-3,-2,0,1,1,3,0,2,1,2,4,1,1,1,2,1,0,2,-1,-1,1] |
| 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^2-4w^3z+23w^2z+31w |
| Inner characteristic polynomial | t^6+40t^4+14t^2+1 |
| Outer characteristic polynomial | t^7+64t^5+59t^3+9t |
| Flat arrow polynomial | 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 288*K1**4*K2 - 1120*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 384*K1**3*K2*K3 - 256*K1**3*K3 - 448*K1**2*K2**4 + 2048*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 6320*K1**2*K2**2 + 160*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 7720*K1**2*K2 - 480*K1**2*K3**2 - 32*K1**2*K4**2 - 5912*K1**2 + 864*K1*K2**3*K3 - 1184*K1*K2**2*K3 + 32*K1*K2*K3**3 - 128*K1*K2*K3*K4 + 6536*K1*K2*K3 + 1152*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1400*K2**4 - 624*K2**2*K3**2 - 16*K2**2*K4**2 + 1104*K2**2*K4 - 3374*K2**2 + 240*K2*K3*K5 + 8*K2*K4*K6 - 1920*K3**2 - 542*K4**2 - 56*K5**2 - 2*K6**2 + 4220 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{5, 6}, {3, 4}, {1, 2}]] |
| If K is slice | False |