# L11a335

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a335 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^3 v^4-3 u^3 v^3+3 u^3 v^2+u^2 v^5-5 u^2 v^4+12 u^2 v^3-11 u^2 v^2+4 u^2 v+4 u v^4-11 u v^3+12 u v^2-5 u v+u+3 v^3-3 v^2+v}{u^{3/2} v^{5/2}}$ (db) Jones polynomial $\frac{26}{q^{9/2}}-\frac{26}{q^{7/2}}+\frac{22}{q^{5/2}}+q^{3/2}-\frac{16}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{11}{q^{15/2}}+\frac{17}{q^{13/2}}-\frac{23}{q^{11/2}}-4 \sqrt{q}+\frac{9}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z^5 a^7+2 z^3 a^7+3 z a^7+2 a^7 z^{-1} -z^7 a^5-3 z^5 a^5-5 z^3 a^5-6 z a^5-3 a^5 z^{-1} -z^7 a^3-3 z^5 a^3-4 z^3 a^3-2 z a^3+a^3 z^{-1} +z^5 a+2 z^3 a+z a$ (db) Kauffman polynomial $-z^5 a^{11}+z^3 a^{11}-4 z^6 a^{10}+3 z^4 a^{10}-10 z^7 a^9+16 z^5 a^9-13 z^3 a^9+6 z a^9-13 z^8 a^8+20 z^6 a^8-10 z^4 a^8+z^2 a^8-10 z^9 a^7+9 z^7 a^7+z^5 a^7+3 z^3 a^7-6 z a^7+2 a^7 z^{-1} -3 z^{10} a^6-18 z^8 a^6+48 z^6 a^6-39 z^4 a^6+15 z^2 a^6-3 a^6-17 z^9 a^5+31 z^7 a^5-20 z^5 a^5+16 z^3 a^5-12 z a^5+3 a^5 z^{-1} -3 z^{10} a^4-12 z^8 a^4+39 z^6 a^4-36 z^4 a^4+17 z^2 a^4-3 a^4-7 z^9 a^3+8 z^7 a^3+5 z^5 a^3-7 z^3 a^3+z a^3+a^3 z^{-1} -7 z^8 a^2+14 z^6 a^2-8 z^4 a^2+2 z^2 a^2-a^2-4 z^7 a+9 z^5 a-6 z^3 a+z a-z^6+2 z^4-z^2$ (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 \
-8-7-6-5-4-3-2-10123χ
4           1-1
2          3 3
0         61 -5
-2        103  7
-4       137   -6
-6      139    4
-8     1313     0
-10    1013      -3
-12   713       6
-14  410        -6
-16  7         7
-1814          -3
-201           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}$ $r=-6$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-4$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-3$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=-2$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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