| Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,0,3,1,2,3,2,0,2,3,1,0,2,1,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.615'] |
| 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+72t^5+256t^3+24t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.615'] |
| 2-strand cable arrow polynomial of the knot is: -80*K1**4 + 224*K1**3*K2*K3 - 32*K1**3*K3 - 192*K1**2*K2**2*K3**2 - 3856*K1**2*K2**2 + 32*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 320*K1**2*K2*K4 + 2368*K1**2*K2 - 320*K1**2*K3**2 - 2384*K1**2 + 1536*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 256*K1*K2**2*K3 - 320*K1*K2**2*K5 + 128*K1*K2*K3**3 - 32*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5528*K1*K2*K3 - 64*K1*K2*K4*K5 + 624*K1*K3*K4 - 824*K2**4 - 1488*K2**2*K3**2 - 112*K2**2*K4**2 + 456*K2**2*K4 - 1468*K2**2 - 160*K2*K3**2*K4 + 528*K2*K3*K5 + 104*K2*K4*K6 - 16*K3**4 + 8*K3**2*K6 - 1732*K3**2 - 242*K4**2 - 12*K5**2 - 4*K6**2 + 2088 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.615'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71569', 'vk6.71680', 'vk6.72098', 'vk6.72308', 'vk6.74034', 'vk6.74596', 'vk6.76080', 'vk6.76792', 'vk6.77187', 'vk6.77286', 'vk6.77487', 'vk6.77649', 'vk6.79022', 'vk6.79600', 'vk6.80560', 'vk6.81011', 'vk6.81105', 'vk6.81136', 'vk6.81159', 'vk6.81216', 'vk6.81311', 'vk6.81455', 'vk6.82251', 'vk6.83506', 'vk6.83835', 'vk6.83973', 'vk6.85403', 'vk6.86315', 'vk6.87114', 'vk6.88010', 'vk6.88329', 'vk6.88963'] |
| 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 | O1O2O3O4U5O6U1O5U2U6U3U4 |
| R3 orbit | {'O1O2O3O4U5O6U1O5U2U6U3U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U2U5U3O6U4O5U6 |
| Gauss code of K* | O1O2O3O4U5U1U3U4O6U2O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6U3O5U1U2U4U6 |
| 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 -2 1 3 0 1],[ 3 0 0 2 3 3 1],[ 2 0 0 2 3 2 0],[-1 -2 -2 0 1 0 -1],[-3 -3 -3 -1 0 -2 -1],[ 0 -3 -2 0 2 0 1],[-1 -1 0 1 1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 1 0 -2 -3],[-3 0 -1 -1 -2 -3 -3],[-1 1 0 1 -1 0 -1],[-1 1 -1 0 0 -2 -2],[ 0 2 1 0 0 -2 -3],[ 2 3 0 2 2 0 0],[ 3 3 1 2 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,0,2,3,1,1,2,3,3,-1,1,0,1,0,2,2,2,3,0] |
| Phi over symmetry | [-3,-2,0,1,1,3,0,3,1,2,3,2,0,2,3,1,0,2,1,1,1] |
| Phi of -K | [-3,-2,0,1,1,3,1,0,2,3,3,0,1,3,2,1,0,1,1,1,1] |
| Phi of K* | [-3,-1,-1,0,2,3,1,1,1,2,3,-1,1,1,2,0,3,3,0,0,1] |
| Phi of -K* | [-3,-2,0,1,1,3,0,3,1,2,3,2,0,2,3,1,0,2,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+6w^3z^2-8w^3z+25w^2z+19w |
| Inner characteristic polynomial | t^6+48t^4+131t^2+9 |
| Outer characteristic polynomial | t^7+72t^5+256t^3+24t |
| Flat arrow polynomial | -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| 2-strand cable arrow polynomial | -80*K1**4 + 224*K1**3*K2*K3 - 32*K1**3*K3 - 192*K1**2*K2**2*K3**2 - 3856*K1**2*K2**2 + 32*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 320*K1**2*K2*K4 + 2368*K1**2*K2 - 320*K1**2*K3**2 - 2384*K1**2 + 1536*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 256*K1*K2**2*K3 - 320*K1*K2**2*K5 + 128*K1*K2*K3**3 - 32*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5528*K1*K2*K3 - 64*K1*K2*K4*K5 + 624*K1*K3*K4 - 824*K2**4 - 1488*K2**2*K3**2 - 112*K2**2*K4**2 + 456*K2**2*K4 - 1468*K2**2 - 160*K2*K3**2*K4 + 528*K2*K3*K5 + 104*K2*K4*K6 - 16*K3**4 + 8*K3**2*K6 - 1732*K3**2 - 242*K4**2 - 12*K5**2 - 4*K6**2 + 2088 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {2, 5}, {1, 3}]] |
| If K is slice | False |