| Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,1,3,1,2,0,3,2,3,0,1,1,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.624'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 6*K1*K2 + K2 + 2*K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.362', '6.624', '6.789', '6.859', '6.882', '6.975', '6.989', '6.1048', '6.1057', '6.1158'] |
| Outer characteristic polynomial of the knot is: t^7+67t^5+275t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.624'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4 + 96*K1**3*K2*K3 - 288*K1**3*K3 + 288*K1**2*K2**3 - 2224*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 32*K1**2*K2*K4 + 4968*K1**2*K2 - 400*K1**2*K3**2 - 32*K1**2*K3*K5 - 5324*K1**2 + 480*K1*K2**3*K3 - 1824*K1*K2**2*K3 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 + 5976*K1*K2*K3 + 968*K1*K3*K4 + 144*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 760*K2**4 - 32*K2**3*K6 - 512*K2**2*K3**2 - 24*K2**2*K4**2 + 1392*K2**2*K4 - 4008*K2**2 - 32*K2*K3**2*K4 + 336*K2*K3*K5 + 32*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 2308*K3**2 - 554*K4**2 - 120*K5**2 - 8*K6**2 + 3928 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.624'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71633', 'vk6.71806', 'vk6.72228', 'vk6.72362', 'vk6.73403', 'vk6.73598', 'vk6.73872', 'vk6.74287', 'vk6.74911', 'vk6.75372', 'vk6.75671', 'vk6.75875', 'vk6.76467', 'vk6.77255', 'vk6.77349', 'vk6.77599', 'vk6.77695', 'vk6.78338', 'vk6.78871', 'vk6.79326', 'vk6.80119', 'vk6.80292', 'vk6.80421', 'vk6.80791', 'vk6.82029', 'vk6.82767', 'vk6.85371', 'vk6.86699', 'vk6.86942', 'vk6.87049', 'vk6.87604', 'vk6.89481'] |
| 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 | O1O2O3O4U5O6U2O5U1U6U4U3 |
| R3 orbit | {'O1O2O3O4U5O6U2O5U1U6U4U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U2U1U5U4O6U3O5U6 |
| Gauss code of K* | O1O2O3O4U1U5U4U3O6U2O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6U3O5U2U1U6U4 |
| 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 2 2 0 1],[ 3 0 1 4 3 2 1],[ 2 -1 0 2 1 2 0],[-2 -4 -2 0 0 -1 -1],[-2 -3 -1 0 0 -1 -1],[ 0 -2 -2 1 1 0 1],[-1 -1 0 1 1 -1 0]] |
| Primitive based matrix | [[ 0 2 2 1 0 -2 -3],[-2 0 0 -1 -1 -1 -3],[-2 0 0 -1 -1 -2 -4],[-1 1 1 0 -1 0 -1],[ 0 1 1 1 0 -2 -2],[ 2 1 2 0 2 0 -1],[ 3 3 4 1 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,0,2,3,0,1,1,1,3,1,1,2,4,1,0,1,2,2,1] |
| Phi over symmetry | [-3,-2,0,1,2,2,0,1,3,1,2,0,3,2,3,0,1,1,0,0,0] |
| Phi of -K | [-3,-2,0,1,2,2,0,1,3,1,2,0,3,2,3,0,1,1,0,0,0] |
| Phi of K* | [-2,-2,-1,0,2,3,0,0,1,2,1,0,1,3,2,0,3,3,0,1,0] |
| Phi of -K* | [-3,-2,0,1,2,2,1,2,1,3,4,2,0,1,2,1,1,1,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 6z^2+25z+27 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+25w^2z+27w |
| Inner characteristic polynomial | t^6+45t^4+134t^2+1 |
| Outer characteristic polynomial | t^7+67t^5+275t^3+6t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 6*K1*K2 + K2 + 2*K3 + 2 |
| 2-strand cable arrow polynomial | -64*K1**4 + 96*K1**3*K2*K3 - 288*K1**3*K3 + 288*K1**2*K2**3 - 2224*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 32*K1**2*K2*K4 + 4968*K1**2*K2 - 400*K1**2*K3**2 - 32*K1**2*K3*K5 - 5324*K1**2 + 480*K1*K2**3*K3 - 1824*K1*K2**2*K3 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 + 5976*K1*K2*K3 + 968*K1*K3*K4 + 144*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 760*K2**4 - 32*K2**3*K6 - 512*K2**2*K3**2 - 24*K2**2*K4**2 + 1392*K2**2*K4 - 4008*K2**2 - 32*K2*K3**2*K4 + 336*K2*K3*K5 + 32*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 2308*K3**2 - 554*K4**2 - 120*K5**2 - 8*K6**2 + 3928 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{2, 6}, {5}, {4}, {3}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {1, 5}, {4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |