# L11a462

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a462 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(3)^3 t(2)^3-t(3)^3 t(2)^3-2 t(1) t(3)^2 t(2)^3+2 t(3)^2 t(2)^3+t(1) t(3) t(2)^3-2 t(1) t(3)^3 t(2)^2+2 t(3)^3 t(2)^2+6 t(1) t(3)^2 t(2)^2-6 t(3)^2 t(2)^2+t(1) t(2)^2-5 t(1) t(3) t(2)^2+4 t(3) t(2)^2-t(3)^3 t(2)-4 t(1) t(3)^2 t(2)+5 t(3)^2 t(2)-2 t(1) t(2)+6 t(1) t(3) t(2)-6 t(3) t(2)+2 t(2)-t(3)^2+t(1)-2 t(1) t(3)+2 t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}$ (db) Jones polynomial $-q+4-7 q^{-1} +13 q^{-2} -17 q^{-3} +21 q^{-4} -21 q^{-5} +19 q^{-6} -14 q^{-7} +10 q^{-8} -4 q^{-9} + q^{-10}$ (db) Signature -4 (db) HOMFLY-PT polynomial $a^8 z^4+2 a^8 z^2+a^8 z^{-2} +2 a^8-2 a^6 z^6-7 a^6 z^4-8 a^6 z^2-2 a^6 z^{-2} -5 a^6+a^4 z^8+5 a^4 z^6+9 a^4 z^4+6 a^4 z^2+a^4 z^{-2} +a^4-a^2 z^6-3 a^2 z^4-a^2 z^2+2 a^2$ (db) Kauffman polynomial $a^{12} z^4+4 a^{11} z^5+10 a^{10} z^6-10 a^{10} z^4+6 a^{10} z^2-2 a^{10}+14 a^9 z^7-18 a^9 z^5+6 a^9 z^3+a^9 z+13 a^8 z^8-17 a^8 z^6+6 a^8 z^4-6 a^8 z^2-a^8 z^{-2} +3 a^8+7 a^7 z^9+4 a^7 z^7-32 a^7 z^5+22 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +2 a^6 z^{10}+16 a^6 z^8-50 a^6 z^6+42 a^6 z^4-22 a^6 z^2-2 a^6 z^{-2} +9 a^6+12 a^5 z^9-26 a^5 z^7+4 a^5 z^5+14 a^5 z^3-8 a^5 z+2 a^5 z^{-1} +2 a^4 z^{10}+7 a^4 z^8-38 a^4 z^6+42 a^4 z^4-14 a^4 z^2-a^4 z^{-2} +3 a^4+5 a^3 z^9-15 a^3 z^7+11 a^3 z^5+a^3 z+4 a^2 z^8-15 a^2 z^6+17 a^2 z^4-4 a^2 z^2-2 a^2+a z^7-3 a z^5+2 a z^3$ (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
3           1-1
1          3 3
-1         41 -3
-3        93  6
-5       106   -4
-7      117    4
-9     1010     0
-11    911      -2
-13   611       5
-15  48        -4
-17  6         6
-1914          -3
-211           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}$ $r=-6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{9}$ $r=-3$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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