# L11n401

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n401 at Knotilus!

### Link Presentations

 Planar diagram presentation X6172 X10,3,11,4 X22,16,19,15 X7,20,8,21 X19,8,20,9 X18,14,5,13 X14,12,15,11 X12,18,13,17 X16,22,17,21 X2536 X4,9,1,10 Gauss code {1, -10, 2, -11}, {-5, 4, 9, -3}, {10, -1, -4, 5, 11, -2, 7, -8, 6, -7, 3, -9, 8, -6}
A Braid Representative
A Morse Link Presentation

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {(v-1)(w-1)\left(uw^{2}-2uw+2u-2w^{2}+2w-1\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}}$ (db) Jones polynomial ${\displaystyle -q^{5}+3q^{4}+3q^{-4}-7q^{3}-5q^{-3}+11q^{2}+11q^{-2}-13q-12q^{-1}+14}$ (db) Signature 0 (db) HOMFLY-PT polynomial ${\displaystyle -z^{2}a^{-4}+2a^{4}z^{-2}+2a^{4}-a^{-4}+a^{2}z^{4}+2z^{4}a^{-2}-2a^{2}z^{2}+3z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-7a^{2}+a^{-2}-z^{6}-2z^{4}+4z^{-2}+5}$ (db) Kauffman polynomial ${\displaystyle az^{9}+z^{9}a^{-1}+4a^{2}z^{8}+4z^{8}a^{-2}+8z^{8}+3a^{3}z^{7}+11az^{7}+13z^{7}a^{-1}+5z^{7}a^{-3}-8a^{2}z^{6}+3z^{6}a^{-4}-11z^{6}-6a^{3}z^{5}-32az^{5}-35z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}+6a^{4}z^{4}+18a^{2}z^{4}-10z^{4}a^{-2}-5z^{4}a^{-4}+7z^{4}+14a^{3}z^{3}+43az^{3}+36z^{3}a^{-1}+5z^{3}a^{-3}-2z^{3}a^{-5}-13a^{4}z^{2}-26a^{2}z^{2}+6z^{2}a^{-2}+3z^{2}a^{-4}-10z^{2}-16a^{3}z-32az-20za^{-1}-3za^{-3}+za^{-5}+9a^{4}+18a^{2}-a^{-4}+11+5a^{3}z^{-1}+9az^{-1}+5a^{-1}z^{-1}+a^{-3}z^{-1}-2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-4z^{-2}}$ (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 \
-4-3-2-1012345χ
11         1-1
9        2 2
7       51 -4
5      62  4
3     75   -2
1    76    1
-1   79     2
-3  45      -1
-5 17       6
-724        -2
-93         3
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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

### 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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