# L11a277

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a277 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1)^3 t(2)^5-t(1)^2 t(2)^5-t(1)^3 t(2)^4+t(1)^2 t(2)^4-t(1) t(2)^4+t(1)^3 t(2)^3-t(1)^2 t(2)^3+t(1) t(2)^3-t(2)^3-t(1)^3 t(2)^2+t(1)^2 t(2)^2-t(1) t(2)^2+t(2)^2-t(1)^2 t(2)+t(1) t(2)-t(2)-t(1)+1}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $\frac{3}{q^{9/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{5/2}}-\frac{1}{q^{3/2}}+\frac{1}{q^{25/2}}-\frac{2}{q^{23/2}}+\frac{3}{q^{21/2}}-\frac{4}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{4}{q^{11/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $a^9 \left(-z^7\right)-6 a^9 z^5-11 a^9 z^3-7 a^9 z-a^9 z^{-1} +a^7 z^9+8 a^7 z^7+23 a^7 z^5+30 a^7 z^3+18 a^7 z+3 a^7 z^{-1} -a^5 z^7-7 a^5 z^5-16 a^5 z^3-13 a^5 z-2 a^5 z^{-1}$ (db) Kauffman polynomial $-z^2 a^{16}-2 z^3 a^{15}-3 z^4 a^{14}+2 z^2 a^{14}-4 z^5 a^{13}+7 z^3 a^{13}-2 z a^{13}-4 z^6 a^{12}+9 z^4 a^{12}-2 z^2 a^{12}-4 z^7 a^{11}+12 z^5 a^{11}-5 z^3 a^{11}-z a^{11}-4 z^8 a^{10}+17 z^6 a^{10}-19 z^4 a^{10}+7 z^2 a^{10}-a^{10}-3 z^9 a^9+15 z^7 a^9-22 z^5 a^9+13 z^3 a^9-6 z a^9+a^9 z^{-1} -z^{10} a^8+2 z^8 a^8+12 z^6 a^8-34 z^4 a^8+23 z^2 a^8-3 a^8-4 z^9 a^7+27 z^7 a^7-61 z^5 a^7+56 z^3 a^7-22 z a^7+3 a^7 z^{-1} -z^{10} a^6+6 z^8 a^6-9 z^6 a^6-3 z^4 a^6+11 z^2 a^6-3 a^6-z^9 a^5+8 z^7 a^5-23 z^5 a^5+29 z^3 a^5-15 z a^5+2 a^5 z^{-1}$ (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 \
-9-8-7-6-5-4-3-2-1012χ
-2           11
-4            0
-6         31 2
-8        11  0
-10       32   1
-12      21    -1
-14     33     0
-16    23      1
-18   22       0
-20  12        1
-22 12         -1
-24 1          1
-261           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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