# L11a317

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a317 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(u^2 v^3-2 u^2 v^2+u^2 v+u v^4-3 u v^3+6 u v^2-3 u v+u+v^3-2 v^2+v\right)}{u^{3/2} v^{5/2}}$ (db) Jones polynomial $-\frac{17}{q^{9/2}}-q^{7/2}+\frac{24}{q^{7/2}}+5 q^{5/2}-\frac{29}{q^{5/2}}-12 q^{3/2}+\frac{28}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{10}{q^{11/2}}+19 \sqrt{q}-\frac{26}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^3 z^7+a z^7-a^5 z^5+3 a^3 z^5+2 a z^5-z^5 a^{-1} -2 a^5 z^3+5 a^3 z^3-z^3 a^{-1} -2 a^5 z+5 a^3 z-3 a z-a^5 z^{-1} +3 a^3 z^{-1} -2 a z^{-1}$ (db) Kauffman polynomial $-4 a^4 z^{10}-4 a^2 z^{10}-9 a^5 z^9-21 a^3 z^9-12 a z^9-8 a^6 z^8-11 a^4 z^8-19 a^2 z^8-16 z^8-4 a^7 z^7+16 a^5 z^7+40 a^3 z^7+8 a z^7-12 z^7 a^{-1} -a^8 z^6+17 a^6 z^6+38 a^4 z^6+48 a^2 z^6-5 z^6 a^{-2} +23 z^6+8 a^7 z^5-11 a^5 z^5-26 a^3 z^5+9 a z^5+15 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-13 a^6 z^4-32 a^4 z^4-29 a^2 z^4+3 z^4 a^{-2} -9 z^4-4 a^7 z^3+7 a^5 z^3+15 a^3 z^3-a z^3-5 z^3 a^{-1} -a^8 z^2+5 a^6 z^2+12 a^4 z^2+8 a^2 z^2+2 z^2-4 a^5 z-9 a^3 z-5 a z-a^6-3 a^4-3 a^2+a^5 z^{-1} +3 a^3 z^{-1} +2 a z^{-1}$ (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 \
-7-6-5-4-3-2-101234χ
8           11
6          4 -4
4         81 7
2        114  -7
0       158   7
-2      1513    -2
-4     1413     1
-6    1015      5
-8   714       -7
-10  310        7
-12 17         -6
-14 3          3
-161           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-3$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-2$ ${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ $r=-1$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15}$ ${\mathbb Z}^{15}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{15}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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