# L11a257

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a257 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1)^2 t(2)^5-t(1) t(2)^5+t(1)^3 t(2)^4-4 t(1)^2 t(2)^4+4 t(1) t(2)^4-t(2)^4-2 t(1)^3 t(2)^3+7 t(1)^2 t(2)^3-6 t(1) t(2)^3+2 t(2)^3+2 t(1)^3 t(2)^2-6 t(1)^2 t(2)^2+7 t(1) t(2)^2-2 t(2)^2-t(1)^3 t(2)+4 t(1)^2 t(2)-4 t(1) t(2)+t(2)-t(1)^2+t(1)}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $\frac{18}{q^{9/2}}-\frac{19}{q^{7/2}}+\frac{16}{q^{5/2}}+q^{3/2}-\frac{13}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{11}{q^{13/2}}-\frac{16}{q^{11/2}}-4 \sqrt{q}+\frac{8}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^7 z^5+3 a^7 z^3+2 a^7 z-a^5 z^7-4 a^5 z^5-6 a^5 z^3-3 a^5 z+a^5 z^{-1} -a^3 z^7-3 a^3 z^5-2 a^3 z^3-a^3 z-a^3 z^{-1} +a z^5+2 a z^3$ (db) Kauffman polynomial $-z^5 a^{11}+2 z^3 a^{11}-3 z^6 a^{10}+6 z^4 a^{10}-2 z^2 a^{10}-5 z^7 a^9+10 z^5 a^9-7 z^3 a^9+2 z a^9-6 z^8 a^8+12 z^6 a^8-11 z^4 a^8+2 z^2 a^8-5 z^9 a^7+8 z^7 a^7-6 z^5 a^7-3 z^3 a^7+z a^7-2 z^{10} a^6-5 z^8 a^6+18 z^6 a^6-21 z^4 a^6+6 z^2 a^6-10 z^9 a^5+24 z^7 a^5-25 z^5 a^5+12 z^3 a^5-a^5 z^{-1} -2 z^{10} a^4-5 z^8 a^4+17 z^6 a^4-10 z^4 a^4+2 z^2 a^4+a^4-5 z^9 a^3+7 z^7 a^3+2 z^5 a^3+z a^3-a^3 z^{-1} -6 z^8 a^2+13 z^6 a^2-4 z^4 a^2-z^2 a^2-4 z^7 a+10 z^5 a-6 z^3 a-z^6+2 z^4-z^2$ (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         51 -4
-2        83  5
-4       96   -3
-6      107    3
-8     89     1
-10    810      -2
-12   49       5
-14  27        -5
-16 14         3
-18 2          -2
-201           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=-3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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