# L11a144

Jump to navigationJump to search

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a144 at Knotilus!

### Link Presentations

 Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X20,14,21,13 X6,19,1,20 X18,11,19,12 X12,6,13,5 X22,16,7,15 X4,18,5,17 X16,22,17,21 Gauss code {1, -2, 3, -10, 8, -6}, {4, -1, 2, -3, 7, -8, 5, -4, 9, -11, 10, -7, 6, -5, 11, -9}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {2u^{2}v^{4}-5u^{2}v^{3}+5u^{2}v^{2}-2u^{2}v-2uv^{4}+7uv^{3}-11uv^{2}+7uv-2u-2v^{3}+5v^{2}-5v+2}{uv^{2}}}}$ (db) Jones polynomial ${\displaystyle 3q^{9/2}-{\frac {4}{q^{9/2}}}-6q^{7/2}+{\frac {8}{q^{7/2}}}+11q^{5/2}-{\frac {13}{q^{5/2}}}-15q^{3/2}+{\frac {16}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+17{\sqrt {q}}-{\frac {19}{\sqrt {q}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle az^{7}+z^{7}a^{-1}-a^{3}z^{5}+3az^{5}+4z^{5}a^{-1}-z^{5}a^{-3}-2a^{3}z^{3}+az^{3}+6z^{3}a^{-1}-3z^{3}a^{-3}-3az+4za^{-1}-2za^{-3}+a^{3}z^{-1}-az^{-1}}$ (db) Kauffman polynomial ${\displaystyle -2z^{10}a^{-2}-2z^{10}-6az^{9}-10z^{9}a^{-1}-4z^{9}a^{-3}-10a^{2}z^{8}-z^{8}a^{-2}-3z^{8}a^{-4}-8z^{8}-11a^{3}z^{7}+2az^{7}+28z^{7}a^{-1}+14z^{7}a^{-3}-z^{7}a^{-5}-8a^{4}z^{6}+13a^{2}z^{6}+18z^{6}a^{-2}+12z^{6}a^{-4}+27z^{6}-4a^{5}z^{5}+15a^{3}z^{5}+14az^{5}-25z^{5}a^{-1}-16z^{5}a^{-3}+4z^{5}a^{-5}-a^{6}z^{4}+7a^{4}z^{4}-2a^{2}z^{4}-22z^{4}a^{-2}-15z^{4}a^{-4}-17z^{4}+2a^{5}z^{3}-7a^{3}z^{3}-8az^{3}+14z^{3}a^{-1}+9z^{3}a^{-3}-4z^{3}a^{-5}-a^{4}z^{2}-a^{2}z^{2}+9z^{2}a^{-2}+6z^{2}a^{-4}+3z^{2}-2az-5za^{-1}-3za^{-3}-a^{2}+a^{3}z^{-1}+az^{-1}}$ (db)

### Khovanov Homology

The coefficients of the monomials ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$).
 \ r \ j \
-5-4-3-2-10123456χ
12           11
10          2 -2
8         41 3
6        72  -5
4       84   4
2      97    -2
0     108     2
-2    710      3
-4   69       -3
-6  38        5
-8 15         -4
-10 3          3
-121           -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.

### Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

Back to the top.