# L10n37

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n37 at Knotilus!

### Link Presentations

 Planar diagram presentation X6172 X12,3,13,4 X13,17,14,16 X9,15,10,14 X15,11,16,10 X17,5,18,20 X7,19,8,18 X19,9,20,8 X2536 X4,11,1,12 Gauss code {1, -9, 2, -10}, {9, -1, -7, 8, -4, 5, 10, -2, -3, 4, -5, 3, -6, 7, -8, 6}
A Braid Representative
A Morse Link Presentation

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {\left(v^{2}-v+1\right)\left(-uv^{2}+uv+u+v^{3}+v^{2}-v\right)}{{\sqrt {u}}v^{5/2}}}}$ (db) Jones polynomial ${\displaystyle -2q^{9/2}+3q^{7/2}-2q^{5/2}-{\frac {1}{q^{5/2}}}+2q^{3/2}-q^{13/2}+2q^{11/2}-2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}}$ (db) Signature 1 (db) HOMFLY-PT polynomial ${\displaystyle -z^{5}a^{-1}+z^{5}a^{-3}+az^{3}-7z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+4az-13za^{-1}+9za^{-3}-2za^{-5}+4az^{-1}-8a^{-1}z^{-1}+5a^{-3}z^{-1}-a^{-5}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle z^{5}a^{-7}-3z^{3}a^{-7}+za^{-7}+2z^{6}a^{-6}-7z^{4}a^{-6}+5z^{2}a^{-6}-2a^{-6}+z^{7}a^{-5}-2z^{5}a^{-5}-2z^{3}a^{-5}+za^{-5}+a^{-5}z^{-1}+3z^{6}a^{-4}-13z^{4}a^{-4}+18z^{2}a^{-4}-9a^{-4}+z^{7}a^{-3}-5z^{5}a^{-3}+9z^{3}a^{-3}-8za^{-3}+5a^{-3}z^{-1}+2z^{6}a^{-2}-14z^{4}a^{-2}+28z^{2}a^{-2}-14a^{-2}+az^{7}+z^{7}a^{-1}-7az^{5}-9z^{5}a^{-1}+15az^{3}+23z^{3}a^{-1}-13az-21za^{-1}+4az^{-1}+8a^{-1}z^{-1}+z^{6}-8z^{4}+15z^{2}-8}$ (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 \
-4-3-2-10123456χ
14          11
12         1 -1
10        11 0
8       21  -1
6     111   -1
4     22    0
2   121     0
0    3      3
-2  1        1
-41          1
-61          1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=0}$ ${\displaystyle i=2}$ ${\displaystyle i=4}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### 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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