| Min(phi) over symmetries of the knot is: [-4,-4,1,2,2,3,0,1,3,4,2,2,4,5,3,1,1,1,0,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.66'] |
| Arrow polynomial of the knot is: -2*K1*K2 + K1 + K3 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.1', '4.3', '6.59', '6.66', '6.112', '6.215', '6.297', '6.306', '6.346', '6.351', '6.352', '6.353', '6.368', '6.393', '6.398', '6.402', '6.420', '6.422', '6.524', '6.529', '6.630', '6.632', '6.633', '6.642', '6.684', '6.707', '6.708', '6.717', '6.719', '6.721', '6.722', '6.737', '6.793', '6.835', '6.837', '6.847', '6.849', '6.857', '6.858', '6.883', '6.902', '6.913', '6.1084', '6.1092', '6.1097', '6.1136', '6.1146', '6.1155', '6.1159', '6.1374', '7.349', '7.365', '7.690', '7.2260', '7.2269', '7.2612', '7.2624', '7.2972', '7.2975', '7.4214', '7.4542', '7.4546', '7.9686', '7.9695', '7.9947', '7.10639', '7.10643', '7.10829', '7.10833', '7.13433', '7.15124', '7.15128', '7.15638', '7.15647', '7.15703', '7.15845', '7.16115', '7.16120', '7.16150', '7.19418', '7.19470', '7.19474', '7.19871', '7.20310', '7.20362', '7.20421', '7.20424', '7.23942', '7.24011', '7.24100', '7.24114', '7.24116', '7.24445', '7.26258', '7.26318', '7.26811', '7.26827', '7.27967', '7.28040', '7.28124', '7.28138', '7.29092', '7.29107', '7.29452', '7.29853', '7.30091', '7.30098', '7.30140', '7.30193', '7.30339', '7.30350', '7.30354'] |
| Outer characteristic polynomial of the knot is: t^7+139t^5+264t^3+13t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.66'] |
| 2-strand cable arrow polynomial of the knot is: -96*K1**3*K3 + 128*K1**2*K2**3 - 2752*K1**2*K2**2 - 192*K1**2*K2*K4 + 3520*K1**2*K2 - 96*K1**2*K3**2 - 32*K1**2*K3*K5 - 3624*K1**2 + 1472*K1*K2**3*K3 - 832*K1*K2**2*K3 + 64*K1*K2**2*K5*K6 - 736*K1*K2**2*K5 - 224*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5072*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K3**2*K5 + 736*K1*K3*K4 + 208*K1*K4*K5 + 64*K1*K5*K6 - 128*K2**6 - 256*K2**4*K3**2 + 192*K2**4*K4 - 1888*K2**4 + 704*K2**3*K3*K5 - 32*K2**3*K6 - 1952*K2**2*K3**2 - 32*K2**2*K3*K7 - 72*K2**2*K4**2 + 1688*K2**2*K4 - 416*K2**2*K5**2 - 32*K2**2*K6**2 - 2262*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1632*K2*K3*K5 + 136*K2*K4*K6 + 80*K2*K5*K7 + 32*K3**2*K6 - 1840*K3**2 - 556*K4**2 - 376*K5**2 - 50*K6**2 + 3002 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.66'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.74171', 'vk6.74175', 'vk6.74773', 'vk6.74781', 'vk6.76315', 'vk6.76323', 'vk6.76840', 'vk6.76844', 'vk6.79203', 'vk6.79205', 'vk6.79669', 'vk6.79673', 'vk6.80683', 'vk6.80687', 'vk6.81051', 'vk6.81053', 'vk6.82893', 'vk6.82922', 'vk6.83417', 'vk6.83425', 'vk6.83436', 'vk6.84012', 'vk6.84027', 'vk6.86290', 'vk6.86680', 'vk6.86682', 'vk6.87518', 'vk6.87526', 'vk6.87862', 'vk6.87909', 'vk6.88611', 'vk6.88619'] |
| 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 | O1O2O3O4O5O6U2U1U5U6U4U3 |
| R3 orbit | {'O1O2O3O4O5O6U2U1U5U6U4U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5O6U4U3U1U2U6U5 |
| Gauss code of K* | O1O2O3O4O5O6U2U1U6U5U3U4 |
| Gauss code of -K* | O1O2O3O4O5O6U3U4U2U1U6U5 |
| 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 -4 -4 2 2 1 3],[ 4 0 0 5 4 2 3],[ 4 0 0 4 3 1 2],[-2 -5 -4 0 0 -1 1],[-2 -4 -3 0 0 -1 1],[-1 -2 -1 1 1 0 1],[-3 -3 -2 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 2 2 1 -4 -4],[-3 0 -1 -1 -1 -2 -3],[-2 1 0 0 -1 -3 -4],[-2 1 0 0 -1 -4 -5],[-1 1 1 1 0 -1 -2],[ 4 2 3 4 1 0 0],[ 4 3 4 5 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-2,-1,4,4,1,1,1,2,3,0,1,3,4,1,4,5,1,2,0] |
| Phi over symmetry | [-4,-4,1,2,2,3,0,1,3,4,2,2,4,5,3,1,1,1,0,1,1] |
| Phi of -K | [-4,-4,1,2,2,3,0,3,1,2,4,4,2,3,5,0,0,1,0,0,0] |
| Phi of K* | [-3,-2,-2,-1,4,4,0,0,1,4,5,0,0,1,2,0,2,3,3,4,0] |
| Phi of -K* | [-4,-4,1,2,2,3,0,1,3,4,2,2,4,5,3,1,1,1,0,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 2t^4-t^3-2t^2-t |
| Normalized Jones-Krushkal polynomial | 8z^2+25z+19 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+12w^3z^2+25w^2z+19w |
| Inner characteristic polynomial | t^6+89t^4+26t^2+1 |
| Outer characteristic polynomial | t^7+139t^5+264t^3+13t |
| Flat arrow polynomial | -2*K1*K2 + K1 + K3 + 1 |
| 2-strand cable arrow polynomial | -96*K1**3*K3 + 128*K1**2*K2**3 - 2752*K1**2*K2**2 - 192*K1**2*K2*K4 + 3520*K1**2*K2 - 96*K1**2*K3**2 - 32*K1**2*K3*K5 - 3624*K1**2 + 1472*K1*K2**3*K3 - 832*K1*K2**2*K3 + 64*K1*K2**2*K5*K6 - 736*K1*K2**2*K5 - 224*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5072*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K3**2*K5 + 736*K1*K3*K4 + 208*K1*K4*K5 + 64*K1*K5*K6 - 128*K2**6 - 256*K2**4*K3**2 + 192*K2**4*K4 - 1888*K2**4 + 704*K2**3*K3*K5 - 32*K2**3*K6 - 1952*K2**2*K3**2 - 32*K2**2*K3*K7 - 72*K2**2*K4**2 + 1688*K2**2*K4 - 416*K2**2*K5**2 - 32*K2**2*K6**2 - 2262*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1632*K2*K3*K5 + 136*K2*K4*K6 + 80*K2*K5*K7 + 32*K3**2*K6 - 1840*K3**2 - 556*K4**2 - 376*K5**2 - 50*K6**2 + 3002 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {3, 5}, {1, 2}]] |
| If K is slice | False |