# L11a386

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a386 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{-3 t(1) t(3)^3+t(1) t(2) t(3)^3-3 t(2) t(3)^3+3 t(3)^3+7 t(1) t(3)^2-4 t(1) t(2) t(3)^2+7 t(2) t(3)^2-5 t(3)^2-7 t(1) t(3)+5 t(1) t(2) t(3)-7 t(2) t(3)+4 t(3)+3 t(1)-3 t(1) t(2)+3 t(2)-1}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db) Jones polynomial $q^{-6} -q^5-2 q^{-5} +4 q^4+7 q^{-4} -10 q^3-11 q^{-3} +15 q^2+19 q^{-2} -20 q-20 q^{-1} +22$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^6 z^{-2} +a^6-3 a^4 z^2-a^4 z^{-2} -z^2 a^{-4} -4 a^4+3 a^2 z^4+2 z^4 a^{-2} +3 a^2 z^2-2 a^2 z^{-2} - a^{-2} z^{-2} -a^2-3 a^{-2} -z^6+4 z^2+3 z^{-2} +7$ (db) Kauffman polynomial $a^2 z^{10}+z^{10}+3 a^3 z^9+8 a z^9+5 z^9 a^{-1} +3 a^4 z^8+12 a^2 z^8+10 z^8 a^{-2} +19 z^8+2 a^5 z^7+a^3 z^7+2 a z^7+12 z^7 a^{-1} +9 z^7 a^{-3} +a^6 z^6-3 a^4 z^6-30 a^2 z^6-13 z^6 a^{-2} +4 z^6 a^{-4} -43 z^6-4 a^5 z^5-12 a^3 z^5-39 a z^5-47 z^5 a^{-1} -15 z^5 a^{-3} +z^5 a^{-5} -4 a^6 z^4-4 a^4 z^4+26 a^2 z^4+4 z^4 a^{-2} -4 z^4 a^{-4} +34 z^4+16 a^3 z^3+53 a z^3+50 z^3 a^{-1} +12 z^3 a^{-3} -z^3 a^{-5} +6 a^6 z^2+6 a^4 z^2-11 a^2 z^2-2 z^2 a^{-2} +z^2 a^{-4} -14 z^2+4 a^5 z-10 a^3 z-34 a z-27 z a^{-1} -7 z a^{-3} -4 a^6-3 a^4+4 a^2+ a^{-2} +5-2 a^5 z^{-1} +2 a^3 z^{-1} +10 a z^{-1} +8 a^{-1} z^{-1} +2 a^{-3} z^{-1} +a^6 z^{-2} +a^4 z^{-2} -2 a^2 z^{-2} - a^{-2} z^{-2} -3 z^{-2}$ (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
11           1-1
9          3 3
7         71 -6
5        83  5
3       127   -5
1      108    2
-1     1113     2
-3    89      -1
-5   311       8
-7  48        -4
-9 16         5
-11 1          -1
-131           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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