# L6a5

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6a5 at Knotilus! L6a5 is ${\displaystyle 6_{1}^{3}}$ in the Rolfsen table of links. It is a closed three-link chain.
 Stained glass window of Trinity symbol, Brazil French coat of arms. Russian coat of arms. Russian passport page-number decoration.

 Planar diagram presentation X6172 X10,3,11,4 X12,7,9,8 X8,11,5,12 X2536 X4,9,1,10 Gauss code {1, -5, 2, -6}, {5, -1, 3, -4}, {6, -2, 4, -3}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {t(2)t(1)+t(3)t(1)-t(1)-t(2)+t(2)t(3)-t(3)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}}}}$ (db) Jones polynomial ${\displaystyle q^{-1}-2q^{-2}+3q^{-3}-q^{-4}+3q^{-5}-q^{-6}+q^{-7}}$ (db) Signature -2 (db) HOMFLY-PT polynomial ${\displaystyle a^{8}z^{-2}-2a^{6}z^{-2}-3a^{6}+2a^{4}z^{2}+a^{4}z^{-2}+3a^{4}+a^{2}z^{2}}$ (db) Kauffman polynomial ${\displaystyle z^{4}a^{8}-3z^{2}a^{8}-a^{8}z^{-2}+3a^{8}+z^{5}a^{7}-z^{3}a^{7}-3za^{7}+2a^{7}z^{-1}+4z^{4}a^{6}-9z^{2}a^{6}-2a^{6}z^{-2}+5a^{6}+z^{5}a^{5}+z^{3}a^{5}-3za^{5}+2a^{5}z^{-1}+3z^{4}a^{4}-5z^{2}a^{4}-a^{4}z^{-2}+3a^{4}+2z^{3}a^{3}+z^{2}a^{2}}$ (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 \
-6-5-4-3-2-10χ
-1      11
-3     21-1
-5    1  1
-7    2  2
-9  31   2
-11 13    2
-13       0
-151      1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

### 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).