# L11n389

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n389 at Knotilus!

### Link Presentations

 Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X13,22,14,19 X9,20,10,21 X19,10,20,11 X21,14,22,15 X11,18,12,5 X15,2,16,3 Gauss code {1, 11, -5, -3}, {-8, 7, -9, 6}, {-4, -1, 2, 5, -7, 8, -10, 4, -6, 9, -11, -2, 3, 10}
A Braid Representative
A Morse Link Presentation

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(t(3)-1)\left(t(1)t(3)^{4}-t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}+t(1)t(3)^{2}-t(1)t(2)t(3)^{2}+t(2)t(3)^{2}-t(3)^{2}-t(2)t(3)+2t(3)+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}}$ (db) Jones polynomial ${\displaystyle q^{-3}-2q^{-4}+5q^{-5}-6q^{-6}+9q^{-7}-7q^{-8}+8q^{-9}-6q^{-10}+3q^{-11}-q^{-12}}$ (db) Signature -6 (db) HOMFLY-PT polynomial ${\displaystyle -a^{14}z^{-2}+z^{2}a^{12}+4a^{12}z^{-2}+6a^{12}-3z^{4}a^{10}-13z^{2}a^{10}-5a^{10}z^{-2}-15a^{10}+2z^{6}a^{8}+9z^{4}a^{8}+12z^{2}a^{8}+2a^{8}z^{-2}+8a^{8}+z^{6}a^{6}+4z^{4}a^{6}+4z^{2}a^{6}+a^{6}}$ (db) Kauffman polynomial ${\displaystyle z^{3}a^{15}-2za^{15}+a^{15}z^{-1}+3z^{4}a^{14}-3z^{2}a^{14}-a^{14}z^{-2}+2a^{14}+2z^{7}a^{13}-7z^{5}a^{13}+18z^{3}a^{13}-15za^{13}+5a^{13}z^{-1}+3z^{8}a^{12}-12z^{6}a^{12}+26z^{4}a^{12}-25z^{2}a^{12}-4a^{12}z^{-2}+14a^{12}+z^{9}a^{11}+3z^{7}a^{11}-22z^{5}a^{11}+44z^{3}a^{11}-33za^{11}+9a^{11}z^{-1}+6z^{8}a^{10}-24z^{6}a^{10}+42z^{4}a^{10}-42z^{2}a^{10}-5a^{10}z^{-2}+21a^{10}+z^{9}a^{9}+3z^{7}a^{9}-21z^{5}a^{9}+30z^{3}a^{9}-20za^{9}+5a^{9}z^{-1}+3z^{8}a^{8}-11z^{6}a^{8}+15z^{4}a^{8}-16z^{2}a^{8}-2a^{8}z^{-2}+9a^{8}+2z^{7}a^{7}-6z^{5}a^{7}+3z^{3}a^{7}+z^{6}a^{6}-4z^{4}a^{6}+4z^{2}a^{6}-a^{6}}$ (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 \
-9-8-7-6-5-4-3-2-10χ
-5         11
-7        21-1
-9       3  3
-11      32  -1
-13     63   3
-15    35    2
-17   54     1
-19  13      2
-21 25       -3
-23 2        2
-251         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-7}$ ${\displaystyle i=-5}$ ${\displaystyle r=-9}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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

### Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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