# L11a375

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a375 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(2)^4 t(1)^4-t(2)^3 t(1)^4-2 t(2)^4 t(1)^3+6 t(2)^3 t(1)^3-5 t(2)^2 t(1)^3+t(2) t(1)^3+t(2)^4 t(1)^2-6 t(2)^3 t(1)^2+11 t(2)^2 t(1)^2-6 t(2) t(1)^2+t(1)^2+t(2)^3 t(1)-5 t(2)^2 t(1)+6 t(2) t(1)-2 t(1)-t(2)+1}{t(1)^2 t(2)^2}$ (db) Jones polynomial $q^{9/2}-3 q^{7/2}+7 q^{5/2}-12 q^{3/2}+15 \sqrt{q}-\frac{19}{\sqrt{q}}+\frac{18}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{12}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-a z^9+a^3 z^7-7 a z^7+z^7 a^{-1} +5 a^3 z^5-19 a z^5+5 z^5 a^{-1} +9 a^3 z^3-24 a z^3+9 z^3 a^{-1} +6 a^3 z-13 a z+6 z a^{-1} +a^3 z^{-1} -a z^{-1}$ (db) Kauffman polynomial $a^7 z^5-2 a^7 z^3+a^7 z+3 a^6 z^6-5 a^6 z^4+2 a^6 z^2+5 a^5 z^7-7 a^5 z^5+2 a^5 z^3+6 a^4 z^8-9 a^4 z^6+z^6 a^{-4} +7 a^4 z^4-3 z^4 a^{-4} -4 a^4 z^2+2 z^2 a^{-4} +5 a^3 z^9-7 a^3 z^7+3 z^7 a^{-3} +9 a^3 z^5-8 z^5 a^{-3} -10 a^3 z^3+5 z^3 a^{-3} +6 a^3 z-z a^{-3} -a^3 z^{-1} +2 a^2 z^{10}+5 a^2 z^8+5 z^8 a^{-2} -17 a^2 z^6-13 z^6 a^{-2} +21 a^2 z^4+10 z^4 a^{-2} -10 a^2 z^2-4 z^2 a^{-2} +a^2+10 a z^9+5 z^9 a^{-1} -27 a z^7-12 z^7 a^{-1} +38 a z^5+13 z^5 a^{-1} -32 a z^3-13 z^3 a^{-1} +14 a z+6 z a^{-1} -a z^{-1} +2 z^{10}+4 z^8-19 z^6+22 z^4-10 z^2$ (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 \
-6-5-4-3-2-1012345χ
10           1-1
8          2 2
6         51 -4
4        72  5
2       85   -3
0      117    4
-2     89     1
-4    810      -2
-6   59       4
-8  27        -5
-10 15         4
-12 2          -2
-141           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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