# L11n379

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n379 at Knotilus!

### Link Presentations

 Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X22,18,19,17 X20,12,21,11 X12,20,13,19 X18,22,5,21 X9,16,10,17 X13,2,14,3 Gauss code {1, 11, -5, -3}, {8, -7, 9, -6}, {-4, -1, 2, 5, -10, 4, 7, -8, -11, -2, 3, 10, 6, -9}
A Braid Representative
A Morse Link Presentation

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {(t(3)-1)(t(3)+1)\left(t(1)t(3)^{3}+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}}$ (db) Jones polynomial ${\displaystyle q^{2}+1+q^{-1}+q^{-2}+q^{-3}-q^{-4}}$ (db) Signature -3 (db) HOMFLY-PT polynomial ${\displaystyle -a^{6}z^{-2}-a^{6}+a^{4}z^{4}+6a^{4}z^{2}+4a^{4}z^{-2}+8a^{4}-a^{2}z^{6}-7a^{2}z^{4}-15a^{2}z^{2}-5a^{2}z^{-2}-13a^{2}+z^{4}+5z^{2}+2z^{-2}+6}$ (db) Kauffman polynomial ${\displaystyle a^{2}z^{8}+z^{8}+a^{3}z^{7}+az^{7}-a^{4}z^{6}-9a^{2}z^{6}-8z^{6}-a^{5}z^{5}-9a^{3}z^{5}-8az^{5}+7a^{4}z^{4}+28a^{2}z^{4}+21z^{4}+6a^{5}z^{3}+26a^{3}z^{3}+20az^{3}-15a^{4}z^{2}-37a^{2}z^{2}-22z^{2}-9a^{5}z-27a^{3}z-18az+a^{6}+12a^{4}+21a^{2}+11+a^{7}z^{-1}+5a^{5}z^{-1}+9a^{3}z^{-1}+5az^{-1}-a^{6}z^{-2}-4a^{4}z^{-2}-5a^{2}z^{-2}-2z^{-2}}$ (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 \
-4-3-2-101234χ
5        11
3        11
1      1  1
-1    2    2
-3    31   2
-5  2      2
-7  11     0
-912       -1
-1111       0
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-5}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

### 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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