# L10a174

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a174 at Knotilus! L10a174 is a closed five-link chain.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v w x+u v w y-u v w+u v x y-u v x-2 u v y+u v+u w x y-2 u w x-u w y+u w-2 u x y+2 u x+2 u y-u+v w x y-2 v w x-2 v w y+2 v w-v x y+v x+2 v y-v-w x y+2 w x+w y-w+x y-x-y}{\sqrt{u} \sqrt{v} \sqrt{w} \sqrt{x} \sqrt{y}}$ (db) Jones polynomial $q^{-12} - q^{-11} +6 q^{-10} -6 q^{-9} +15 q^{-8} -11 q^{-7} +15 q^{-6} -10 q^{-5} +10 q^{-4} -4 q^{-3} + q^{-2}$ (db) Signature -4 (db) HOMFLY-PT polynomial $a^{14} z^{-4} -4 a^{12} z^{-4} -5 a^{12} z^{-2} +6 a^{10} z^{-4} +15 a^{10} z^{-2} +10 a^{10}-4 a^8 z^{-4} -10 a^8 z^2-15 a^8 z^{-2} -20 a^8+4 a^6 z^4+a^6 z^{-4} +10 a^6 z^2+5 a^6 z^{-2} +10 a^6+a^4 z^4$ (db) Kauffman polynomial $a^{14} z^6-5 a^{14} z^4-a^{14} z^{-4} +10 a^{14} z^2+5 a^{14} z^{-2} -10 a^{14}+a^{13} z^7-10 a^{13} z^3+4 a^{13} z^{-3} +20 a^{13} z-15 a^{13} z^{-1} +a^{12} z^8+4 a^{12} z^6-20 a^{12} z^4-4 a^{12} z^{-4} +30 a^{12} z^2+14 a^{12} z^{-2} -25 a^{12}+a^{11} z^9+2 a^{11} z^7+2 a^{11} z^5-30 a^{11} z^3+12 a^{11} z^{-3} +55 a^{11} z-41 a^{11} z^{-1} +6 a^{10} z^8-2 a^{10} z^6-25 a^{10} z^4-6 a^{10} z^{-4} +40 a^{10} z^2+18 a^{10} z^{-2} -31 a^{10}+a^9 z^9+11 a^9 z^7-12 a^9 z^5-30 a^9 z^3+12 a^9 z^{-3} +55 a^9 z-41 a^9 z^{-1} +5 a^8 z^8+5 a^8 z^6-25 a^8 z^4-4 a^8 z^{-4} +30 a^8 z^2+14 a^8 z^{-2} -25 a^8+10 a^7 z^7-10 a^7 z^5-10 a^7 z^3+4 a^7 z^{-3} +20 a^7 z-15 a^7 z^{-1} +10 a^6 z^6-14 a^6 z^4-a^6 z^{-4} +10 a^6 z^2+5 a^6 z^{-2} -10 a^6+4 a^5 z^5+a^4 z^4$ (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 \
-10-9-8-7-6-5-4-3-2-10χ
-3          11
-5         41-3
-7        6  6
-9       44  0
-11      116   5
-13     1014    4
-15    51     4
-17   110      9
-19  55       0
-21 16        5
-23           0
-251          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{6}$ ${\mathbb Z}^{5}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-4$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{11}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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