# L11n339

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n339 at Knotilus!

 Planar diagram presentation X6172 X10,3,11,4 X18,12,19,11 X7,14,8,15 X13,8,14,9 X19,22,20,13 X15,20,16,21 X21,16,22,17 X12,18,5,17 X2536 X4,9,1,10 Gauss code {1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, 3, -9}, {-5, 4, -7, 8, 9, -3, -6, 7, -8, 6}
A Braid Representative
A Morse Link Presentation

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {(t(2)-1)(t(3)-1)\left(t(3)t(2)^{2}+t(1)t(3)^{2}t(2)-t(3)^{2}t(2)+t(1)t(2)-t(2)-t(1)t(3)\right)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}}$ (db) Jones polynomial ${\displaystyle 1-2q^{-1}+4q^{-2}-3q^{-3}+4q^{-4}-2q^{-5}+3q^{-6}-q^{-8}+q^{-9}-q^{-10}}$ (db) Signature -4 (db) HOMFLY-PT polynomial ${\displaystyle -a^{10}z^{-2}-a^{10}+2z^{2}a^{8}+4a^{8}z^{-2}+5a^{8}-2z^{2}a^{6}-5a^{6}z^{-2}-7a^{6}-z^{6}a^{4}-4z^{4}a^{4}-3z^{2}a^{4}+2a^{4}z^{-2}+a^{4}+z^{4}a^{2}+3z^{2}a^{2}+2a^{2}}$ (db) Kauffman polynomial ${\displaystyle z^{7}a^{11}-6z^{5}a^{11}+10z^{3}a^{11}-6za^{11}+a^{11}z^{-1}+z^{8}a^{10}-6z^{6}a^{10}+9z^{4}a^{10}-6z^{2}a^{10}-a^{10}z^{-2}+4a^{10}+2z^{7}a^{9}-16z^{5}a^{9}+33z^{3}a^{9}-22za^{9}+5a^{9}z^{-1}+z^{8}a^{8}-9z^{6}a^{8}+22z^{4}a^{8}-22z^{2}a^{8}-4a^{8}z^{-2}+15a^{8}+2z^{7}a^{7}-15z^{5}a^{7}+34z^{3}a^{7}-30za^{7}+9a^{7}z^{-1}+z^{8}a^{6}-5z^{6}a^{6}+11z^{4}a^{6}-19z^{2}a^{6}-5a^{6}z^{-2}+16a^{6}+3z^{7}a^{5}-12z^{5}a^{5}+15z^{3}a^{5}-13za^{5}+5a^{5}z^{-1}+z^{8}a^{4}-z^{6}a^{4}-6z^{4}a^{4}+2z^{2}a^{4}-2a^{4}z^{-2}+4a^{4}+2z^{7}a^{3}-7z^{5}a^{3}+4z^{3}a^{3}+za^{3}+z^{6}a^{2}-4z^{4}a^{2}+5z^{2}a^{2}-2a^{2}}$ (db)

### Khovanov Homology

The coefficients of the monomials ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$).
 \ r \ j \
-9-8-7-6-5-4-3-2-1012χ
1           11
-1          1 -1
-3         31 2
-5        23  1
-7      131   1
-9     112    2
-11    153     1
-13   1 2      3
-15   12       -1
-17 11         0
-19            0
-211           -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-5}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-9}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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