# L7a7

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a7 at Knotilus! L7a7 is ${\displaystyle 7_{1}^{3}}$ in the Rolfsen table of links.

 Planar diagram presentation X6172 X10,3,11,4 X14,12,9,11 X8,14,5,13 X12,8,13,7 X2536 X4,9,1,10 Gauss code {1, -6, 2, -7}, {6, -1, 5, -4}, {7, -2, 3, -5, 4, -3}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {uvw+u(-v)-uw+2u-2vw+v+w-1}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}}}}$ (db) Jones polynomial ${\displaystyle q^{-4}-q^{3}-q^{-3}+3q^{2}+4q^{-2}-3q-3q^{-1}+4}$ (db) Signature 0 (db) HOMFLY-PT polynomial ${\displaystyle a^{4}z^{-2}+a^{4}-2z^{2}a^{2}-2a^{2}z^{-2}-3a^{2}+z^{4}+2z^{2}+z^{-2}+2-z^{2}a^{-2}}$ (db) Kauffman polynomial ${\displaystyle a^{2}z^{6}+z^{6}+a^{3}z^{5}+4az^{5}+3z^{5}a^{-1}+a^{4}z^{4}+a^{2}z^{4}+3z^{4}a^{-2}+3z^{4}-4az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}-3a^{4}z^{2}-5a^{2}z^{2}-3z^{2}a^{-2}-5z^{2}-3a^{3}z-3az+3a^{4}+5a^{2}+3+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}}$ (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 \
-4-3-2-10123χ
7       1-1
5      2 2
3     11 0
1    32  1
-1   34   1
-3  1     1
-5  3     3
-711      0
-91       1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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