# L11a542

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a542 at Knotilus!

 Planar diagram presentation X6172 X12,6,13,5 X8493 X2,16,3,15 X16,7,17,8 X22,20,15,19 X14,22,11,21 X20,14,21,13 X18,10,19,9 X10,12,5,11 X4,17,1,18 Gauss code {1, -4, 3, -11}, {10, -2, 8, -7}, {2, -1, 5, -3, 9, -10}, {4, -5, 11, -9, 6, -8, 7, -6}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(u-1)(w-1)(x-1)(2vwx+v(-w)-vx+v-wx+w+x-2)}{{\sqrt {u}}{\sqrt {v}}wx}}}$ (db) Jones polynomial ${\displaystyle 25q^{9/2}-27q^{7/2}+20q^{5/2}-17q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+5q^{17/2}-11q^{15/2}+17q^{13/2}-24q^{11/2}+8{\sqrt {q}}-{\frac {4}{\sqrt {q}}}}$ (db) Signature 3 (db) HOMFLY-PT polynomial ${\displaystyle z^{7}a^{-3}+z^{7}a^{-5}-z^{5}a^{-1}+3z^{5}a^{-3}+2z^{5}a^{-5}-z^{5}a^{-7}-2z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-z^{3}a^{-7}+2za^{-3}-3za^{-5}+za^{-7}+2a^{-1}z^{-1}-4a^{-3}z^{-1}+2a^{-5}z^{-1}+a^{-1}z^{-3}-3a^{-3}z^{-3}+3a^{-5}z^{-3}-a^{-7}z^{-3}}$ (db) Kauffman polynomial ${\displaystyle z^{5}a^{-11}+5z^{6}a^{-10}-4z^{4}a^{-10}+11z^{7}a^{-9}-15z^{5}a^{-9}+6z^{3}a^{-9}+13z^{8}a^{-8}-16z^{6}a^{-8}+3z^{4}a^{-8}+z^{2}a^{-8}+8z^{9}a^{-7}+6z^{7}a^{-7}-31z^{5}a^{-7}+20z^{3}a^{-7}+a^{-7}z^{-3}-3za^{-7}-2a^{-7}z^{-1}+2z^{10}a^{-6}+22z^{8}a^{-6}-43z^{6}a^{-6}+14z^{4}a^{-6}+3z^{2}a^{-6}-3a^{-6}z^{-2}+4a^{-6}+13z^{9}a^{-5}-5z^{7}a^{-5}-35z^{5}a^{-5}+36z^{3}a^{-5}+3a^{-5}z^{-3}-11za^{-5}-3a^{-5}z^{-1}+2z^{10}a^{-4}+15z^{8}a^{-4}-32z^{6}a^{-4}+9z^{4}a^{-4}+4z^{2}a^{-4}-6a^{-4}z^{-2}+7a^{-4}+5z^{9}a^{-3}+4z^{7}a^{-3}-30z^{5}a^{-3}+32z^{3}a^{-3}+3a^{-3}z^{-3}-11za^{-3}-3a^{-3}z^{-1}+6z^{8}a^{-2}-9z^{6}a^{-2}+3z^{2}a^{-2}-3a^{-2}z^{-2}+4a^{-2}+4z^{7}a^{-1}-10z^{5}a^{-1}+10z^{3}a^{-1}+a^{-1}z^{-3}-3za^{-1}-2a^{-1}z^{-1}+z^{6}-2z^{4}+z^{2}}$ (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 \
-3-2-1012345678χ
20           11
18          4 -4
16         71 6
14        104  -6
12       147   7
10      1312    -1
8     1412     2
6    1017      7
4   710       -3
2  312        9
0 15         -4
-2 3          3
-41           -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=2}$ ${\displaystyle i=4}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{17}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{14}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}}$ ${\displaystyle {\mathbb {Z} }^{13}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}}$ ${\displaystyle {\mathbb {Z} }^{14}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=8}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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

Read me first: Modifying Knot Pages

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