| Min(phi) over symmetries of the knot is: [-4,-2,-1,2,2,3,0,2,2,3,4,1,1,2,2,1,2,2,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.161'] |
| Arrow polynomial of the knot is: -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.58', '6.76', '6.78', '6.90', '6.98', '6.154', '6.161', '6.162', '6.198', '6.280', '6.284', '6.345', '6.417', '6.421', '6.435', '6.511'] |
| Outer characteristic polynomial of the knot is: t^7+113t^5+65t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.161'] |
| 2-strand cable arrow polynomial of the knot is: -144*K1**4 - 416*K1**3*K3 - 64*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 2000*K1**2*K2 - 176*K1**2*K3**2 - 80*K1**2*K4**2 - 2524*K1**2 - 384*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 2120*K1*K2*K3 - 32*K1*K2*K4*K5 + 912*K1*K3*K4 + 152*K1*K4*K5 + 56*K1*K5*K6 - 32*K2**4 - 32*K2**3*K6 - 144*K2**2*K3**2 - 8*K2**2*K4**2 + 608*K2**2*K4 - 8*K2**2*K6**2 - 1970*K2**2 + 264*K2*K3*K5 + 96*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 + 24*K3**2*K6 - 1148*K3**2 - 594*K4**2 - 168*K5**2 - 70*K6**2 - 8*K7**2 - 2*K8**2 + 2002 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.161'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73332', 'vk6.73344', 'vk6.73493', 'vk6.73505', 'vk6.75252', 'vk6.75268', 'vk6.75500', 'vk6.75512', 'vk6.78216', 'vk6.78228', 'vk6.78454', 'vk6.78470', 'vk6.80040', 'vk6.80052', 'vk6.80188', 'vk6.80200', 'vk6.81587', 'vk6.81669', 'vk6.81673', 'vk6.82098', 'vk6.82191', 'vk6.82271', 'vk6.82277', 'vk6.82667', 'vk6.82669', 'vk6.84159', 'vk6.84652', 'vk6.84734', 'vk6.84737', 'vk6.84966', 'vk6.84969', 'vk6.85749', 'vk6.85957', 'vk6.85964', 'vk6.87329', 'vk6.87689', 'vk6.88127', 'vk6.89719', 'vk6.90040', 'vk6.90082'] |
| 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 | O1O2O3O4O5U1O6U4U2U5U6U3 |
| R3 orbit | {'O1O2O3O4O5U1U3O6U2U5U4U6', 'O1O2O3O4O5U1O6U4U2U5U6U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U3U6U1U4U2O6U5 |
| Gauss code of K* | O1O2O3O4O5U6U2U5U1U3O6U4 |
| Gauss code of -K* | O1O2O3O4O5U2O6U3U5U1U4U6 |
| 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 -2 2 -1 2 3],[ 4 0 2 4 1 3 3],[ 2 -2 0 3 0 2 3],[-2 -4 -3 0 -2 0 2],[ 1 -1 0 2 0 1 2],[-2 -3 -2 0 -1 0 1],[-3 -3 -3 -2 -2 -1 0]] |
| Primitive based matrix | [[ 0 3 2 2 -1 -2 -4],[-3 0 -1 -2 -2 -3 -3],[-2 1 0 0 -1 -2 -3],[-2 2 0 0 -2 -3 -4],[ 1 2 1 2 0 0 -1],[ 2 3 2 3 0 0 -2],[ 4 3 3 4 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-2,1,2,4,1,2,2,3,3,0,1,2,3,2,3,4,0,1,2] |
| Phi over symmetry | [-4,-2,-1,2,2,3,0,2,2,3,4,1,1,2,2,1,2,2,0,-1,0] |
| Phi of -K | [-4,-2,-1,2,2,3,0,2,2,3,4,1,1,2,2,1,2,2,0,-1,0] |
| Phi of K* | [-3,-2,-2,1,2,4,-1,0,2,2,4,0,1,1,2,2,2,3,1,2,0] |
| Phi of -K* | [-4,-2,-1,2,2,3,2,1,3,4,3,0,2,3,3,1,2,2,0,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^3-t^2+t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+75t^4+20t^2 |
| Outer characteristic polynomial | t^7+113t^5+65t^3+3t |
| Flat arrow polynomial | -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1 |
| 2-strand cable arrow polynomial | -144*K1**4 - 416*K1**3*K3 - 64*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 2000*K1**2*K2 - 176*K1**2*K3**2 - 80*K1**2*K4**2 - 2524*K1**2 - 384*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 2120*K1*K2*K3 - 32*K1*K2*K4*K5 + 912*K1*K3*K4 + 152*K1*K4*K5 + 56*K1*K5*K6 - 32*K2**4 - 32*K2**3*K6 - 144*K2**2*K3**2 - 8*K2**2*K4**2 + 608*K2**2*K4 - 8*K2**2*K6**2 - 1970*K2**2 + 264*K2*K3*K5 + 96*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 + 24*K3**2*K6 - 1148*K3**2 - 594*K4**2 - 168*K5**2 - 70*K6**2 - 8*K7**2 - 2*K8**2 + 2002 |
| 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}], [{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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}, {2, 5}, {4}, {3}, {1}], [{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}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {3}, {1, 2}]] |
| If K is slice | False |