| Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,-1,2,1,2,4,2,1,1,2,0,0,2,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.322'] |
| 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+61t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.322'] |
| 2-strand cable arrow polynomial of the knot is: -16*K1**4 + 64*K1**3*K2*K3 - 64*K1**2*K2**2*K3**2 - 432*K1**2*K2**2 + 184*K1**2*K2 - 192*K1**2*K3**2 - 496*K1**2 + 224*K1*K2**3*K3 + 96*K1*K2*K3**3 + 1200*K1*K2*K3 + 144*K1*K3*K4 + 24*K1*K5*K6 - 120*K2**4 - 368*K2**2*K3**2 - 16*K2**2*K4**2 + 32*K2**2*K4 - 396*K2**2 + 184*K2*K3*K5 + 24*K2*K4*K6 - 48*K3**4 + 8*K3**2*K6 - 468*K3**2 - 50*K4**2 - 44*K5**2 - 20*K6**2 + 544 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.322'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71567', 'vk6.71646', 'vk6.71649', 'vk6.71652', 'vk6.71668', 'vk6.71823', 'vk6.71826', 'vk6.72094', 'vk6.72243', 'vk6.72247', 'vk6.72250', 'vk6.72267', 'vk6.72371', 'vk6.72377', 'vk6.76076', 'vk6.76789', 'vk6.77185', 'vk6.77262', 'vk6.77268', 'vk6.77358', 'vk6.77366', 'vk6.77378', 'vk6.77484', 'vk6.77606', 'vk6.77616', 'vk6.77698', 'vk6.77706', 'vk6.77717', 'vk6.79598', 'vk6.80558', 'vk6.81009', 'vk6.81146', 'vk6.81302', 'vk6.81406', 'vk6.81408', 'vk6.82461', 'vk6.83975', 'vk6.84441', 'vk6.86324', 'vk6.86949', 'vk6.87094', 'vk6.87153', 'vk6.87163', 'vk6.87785', 'vk6.88003', 'vk6.88019', 'vk6.88120', 'vk6.88333'] |
| 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 | O1O2O3O4O5U2U4O6U5U1U3U6 |
| R3 orbit | {'O1O2O3O4U1U3O5O6U2U4U5U6', 'O1O2O3O4U1O5U4O6U2U5U3U6', 'O1O2O3O4O5U2U4O6U5U1U3U6'} |
| R3 orbit length | 3 |
| Gauss code of -K | O1O2O3O4O5U6U3U5U1O6U2U4 |
| Gauss code of K* | O1O2O3O4U2U5U3U6U1O5O6U4 |
| Gauss code of -K* | O1O2O3O4U1O5O6U4U5U2U6U3 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -2 -3 1 0 1 3],[ 2 0 -2 2 0 2 3],[ 3 2 0 3 1 2 2],[-1 -2 -3 0 -1 1 2],[ 0 0 -1 1 0 1 1],[-1 -2 -2 -1 -1 0 1],[-3 -3 -2 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 1 0 -2 -3],[-3 0 -1 -2 -1 -3 -2],[-1 1 0 -1 -1 -2 -2],[-1 2 1 0 -1 -2 -3],[ 0 1 1 1 0 0 -1],[ 2 3 2 2 0 0 -2],[ 3 2 2 3 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,0,2,3,1,2,1,3,2,1,1,2,2,1,2,3,0,1,2] |
| Phi over symmetry | [-3,-2,0,1,1,3,-1,2,1,2,4,2,1,1,2,0,0,2,-1,0,1] |
| Phi of -K | [-3,-2,0,1,1,3,-1,2,1,2,4,2,1,1,2,0,0,2,-1,0,1] |
| Phi of K* | [-3,-1,-1,0,2,3,0,1,2,2,4,1,0,1,1,0,1,2,2,2,-1] |
| Phi of -K* | [-3,-2,0,1,1,3,2,1,2,3,2,0,2,2,3,1,1,1,-1,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 5z+11 |
| Enhanced Jones-Krushkal polynomial | -4w^3z+9w^2z+11w |
| Inner characteristic polynomial | t^6+48t^4+8t^2 |
| Outer characteristic polynomial | t^7+72t^5+61t^3 |
| Flat arrow polynomial | -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| 2-strand cable arrow polynomial | -16*K1**4 + 64*K1**3*K2*K3 - 64*K1**2*K2**2*K3**2 - 432*K1**2*K2**2 + 184*K1**2*K2 - 192*K1**2*K3**2 - 496*K1**2 + 224*K1*K2**3*K3 + 96*K1*K2*K3**3 + 1200*K1*K2*K3 + 144*K1*K3*K4 + 24*K1*K5*K6 - 120*K2**4 - 368*K2**2*K3**2 - 16*K2**2*K4**2 + 32*K2**2*K4 - 396*K2**2 + 184*K2*K3*K5 + 24*K2*K4*K6 - 48*K3**4 + 8*K3**2*K6 - 468*K3**2 - 50*K4**2 - 44*K5**2 - 20*K6**2 + 544 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {4, 5}, {3}, {1, 2}]] |
| If K is slice | False |