# L11a70

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a70 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1) t(2)^5-2 t(2)^5-5 t(1) t(2)^4+7 t(2)^4+10 t(1) t(2)^3-10 t(2)^3-10 t(1) t(2)^2+10 t(2)^2+7 t(1) t(2)-5 t(2)-2 t(1)+1}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $\frac{22}{q^{9/2}}-\frac{23}{q^{7/2}}+\frac{20}{q^{5/2}}+q^{3/2}-\frac{16}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{13}{q^{13/2}}-\frac{19}{q^{11/2}}-5 \sqrt{q}+\frac{10}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z a^9+a^9 z^{-1} -3 z^3 a^7-6 z a^7-3 a^7 z^{-1} +3 z^5 a^5+8 z^3 a^5+8 z a^5+4 a^5 z^{-1} -z^7 a^3-3 z^5 a^3-4 z^3 a^3-4 z a^3-2 a^3 z^{-1} +z^5 a+z^3 a-z a$ (db) Kauffman polynomial $a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-5 a^{10} z^4+3 a^{10} z^2-a^{10}+5 a^9 z^7-6 a^9 z^5+3 a^9 z^3-2 a^9 z+a^9 z^{-1} +6 a^8 z^8-4 a^8 z^6-4 a^8 z^4+8 a^8 z^2-3 a^8+5 a^7 z^9+2 a^7 z^7-17 a^7 z^5+24 a^7 z^3-13 a^7 z+3 a^7 z^{-1} +2 a^6 z^{10}+12 a^6 z^8-28 a^6 z^6+16 a^6 z^4+4 a^6 z^2-3 a^6+12 a^5 z^9-14 a^5 z^7-15 a^5 z^5+30 a^5 z^3-18 a^5 z+4 a^5 z^{-1} +2 a^4 z^{10}+15 a^4 z^8-42 a^4 z^6+26 a^4 z^4-2 a^4 z^2-2 a^4+7 a^3 z^9-6 a^3 z^7-15 a^3 z^5+15 a^3 z^3-7 a^3 z+2 a^3 z^{-1} +9 a^2 z^8-20 a^2 z^6+10 a^2 z^4-a^2 z^2+5 a z^7-10 a z^5+4 a z^3+a z+z^6-z^4$ (db)

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
-8-7-6-5-4-3-2-10123χ
4           1-1
2          4 4
0         61 -5
-2        104  6
-4       117   -4
-6      129    3
-8     1011     1
-10    912      -3
-12   511       6
-14  28        -6
-16 15         4
-18 2          -2
-201           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{9}$ $r=-3$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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