# L11a436

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a436 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(v-1) (w-1) \left(2 u v w^2-u v w+2 u v-u w^2-u+v^2 w^2+v^2-2 v w^2+v w-2 v\right)}{\sqrt{u} v^{3/2} w^{3/2}}$ (db) Jones polynomial $- q^{-11} +3 q^{-10} -8 q^{-9} +13 q^{-8} -16 q^{-7} +19 q^{-6} -17 q^{-5} +16 q^{-4} -10 q^{-3} +6 q^{-2} -2 q^{-1} +1$ (db) Signature -4 (db) HOMFLY-PT polynomial $a^{10} \left(-z^2\right)-a^{10} z^{-2} -2 a^{10}+3 a^8 z^4+8 a^8 z^2+4 a^8 z^{-2} +8 a^8-2 a^6 z^6-7 a^6 z^4-10 a^6 z^2-5 a^6 z^{-2} -10 a^6-a^4 z^6-2 a^4 z^4+2 a^4 z^{-2} +2 a^4+a^2 z^4+3 a^2 z^2+2 a^2$ (db) Kauffman polynomial $z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-4 z^4 a^{12}+z^2 a^{12}+6 z^7 a^{11}-10 z^5 a^{11}+8 z^3 a^{11}-5 z a^{11}+a^{11} z^{-1} +7 z^8 a^{10}-11 z^6 a^{10}+10 z^4 a^{10}-11 z^2 a^{10}-a^{10} z^{-2} +6 a^{10}+4 z^9 a^9+5 z^7 a^9-29 z^5 a^9+41 z^3 a^9-25 z a^9+5 a^9 z^{-1} +z^{10} a^8+13 z^8 a^8-39 z^6 a^8+55 z^4 a^8-45 z^2 a^8-4 a^8 z^{-2} +21 a^8+7 z^9 a^7-6 z^7 a^7-18 z^5 a^7+43 z^3 a^7-35 z a^7+9 a^7 z^{-1} +z^{10} a^6+9 z^8 a^6-32 z^6 a^6+49 z^4 a^6-44 z^2 a^6-5 a^6 z^{-2} +22 a^6+3 z^9 a^5-3 z^7 a^5-5 z^5 a^5+14 z^3 a^5-15 z a^5+5 a^5 z^{-1} +3 z^8 a^4-6 z^6 a^4+4 z^4 a^4-6 z^2 a^4-2 a^4 z^{-2} +6 a^4+2 z^7 a^3-5 z^5 a^3+2 z^3 a^3+z a^3+z^6 a^2-4 z^4 a^2+5 z^2 a^2-2 a^2$ (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 \
-9-8-7-6-5-4-3-2-1012χ
1           11
-1          1 -1
-3         51 4
-5        73  -4
-7       93   6
-9      87    -1
-11     119     2
-13    710      3
-15   69       -3
-17  27        5
-19 16         -5
-21 2          2
-231           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-5$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{11}$ $r=-3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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