# L10n57

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n57 at Knotilus!

### Link Presentations

 Planar diagram presentation X10,1,11,2 X20,5,9,6 X3,15,4,14 X15,5,16,4 X7,17,8,16 X11,18,12,19 X17,12,18,13 X2,9,3,10 X13,1,14,8 X6,19,7,20 Gauss code {1, -8, -3, 4, 2, -10, -5, 9}, {8, -1, -6, 7, -9, 3, -4, 5, -7, 6, 10, -2}
A Braid Representative
A Morse Link Presentation

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(u-1)(u+1)^{2}(v-1)}{u^{3/2}{\sqrt {v}}}}}$ (db) Jones polynomial ${\displaystyle {\frac {1}{q^{9/2}}}-q^{7/2}-{\frac {1}{q^{7/2}}}+q^{5/2}-{\frac {1}{q^{3/2}}}-{\frac {1}{\sqrt {q}}}}$ (db) Signature 0 (db) HOMFLY-PT polynomial ${\displaystyle -a^{3}z^{3}-3a^{3}z-za^{-3}+az^{5}+5az^{3}+5az-za^{-1}+az^{-1}-a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -a^{3}z^{7}-az^{7}-a^{4}z^{6}-a^{2}z^{6}-z^{6}a^{-2}-z^{6}+6a^{3}z^{5}+7az^{5}-z^{5}a^{-3}+5a^{4}z^{4}+6a^{2}z^{4}+5z^{4}a^{-2}+6z^{4}-10a^{3}z^{3}-14az^{3}+4z^{3}a^{-3}-5a^{4}z^{2}-7a^{2}z^{2}-5z^{2}a^{-2}-7z^{2}+6a^{3}z+10az+2za^{-1}-2za^{-3}+1-az^{-1}-a^{-1}z^{-1}}$ (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 \
-5-4-3-2-101234χ
8         11
6          0
4      111 -1
2     11   0
0    131   1
-2   112    2
-4   1      1
-6 111      1
-8          0
-101         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle i=2}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### 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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