# L10a104

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a104 at Knotilus!

### Link Presentations

 Planar diagram presentation X10,1,11,2 X16,11,17,12 X8,9,1,10 X20,17,9,18 X12,4,13,3 X14,8,15,7 X6,14,7,13 X18,6,19,5 X4,20,5,19 X2,16,3,15 Gauss code {1, -10, 5, -9, 8, -7, 6, -3}, {3, -1, 2, -5, 7, -6, 10, -2, 4, -8, 9, -4}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {u^{3}\left(-v^{3}\right)+2u^{3}v^{2}-u^{3}v+3u^{2}v^{3}-7u^{2}v^{2}+7u^{2}v-2u^{2}-2uv^{3}+7uv^{2}-7uv+3u-v^{2}+2v-1}{u^{3/2}v^{3/2}}}}$ (db) Jones polynomial ${\displaystyle -7q^{9/2}+11q^{7/2}-{\frac {1}{q^{7/2}}}-15q^{5/2}+{\frac {4}{q^{5/2}}}+15q^{3/2}-{\frac {8}{q^{3/2}}}-q^{13/2}+3q^{11/2}-15{\sqrt {q}}+{\frac {12}{\sqrt {q}}}}$ (db) Signature 1 (db) HOMFLY-PT polynomial ${\displaystyle -z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+2z^{5}a^{-3}+6z^{3}a^{-3}+6za^{-3}+a^{-3}z^{-1}-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+2az^{3}-6z^{3}a^{-1}+az-3za^{-1}}$ (db) Kauffman polynomial ${\displaystyle z^{5}a^{-7}-2z^{3}a^{-7}+za^{-7}+3z^{6}a^{-6}-5z^{4}a^{-6}+2z^{2}a^{-6}+5z^{7}a^{-5}-8z^{5}a^{-5}+6z^{3}a^{-5}-4za^{-5}+a^{-5}z^{-1}+5z^{8}a^{-4}-5z^{6}a^{-4}+2z^{2}a^{-4}-a^{-4}+2z^{9}a^{-3}+9z^{7}a^{-3}+a^{3}z^{5}-25z^{5}a^{-3}-a^{3}z^{3}+23z^{3}a^{-3}-9za^{-3}+a^{-3}z^{-1}+11z^{8}a^{-2}+4a^{2}z^{6}-17z^{6}a^{-2}-6a^{2}z^{4}+6z^{4}a^{-2}+2a^{2}z^{2}+z^{2}a^{-2}+2z^{9}a^{-1}+7az^{7}+11z^{7}a^{-1}-12az^{5}-29z^{5}a^{-1}+6az^{3}+22z^{3}a^{-1}-2az-6za^{-1}+6z^{8}-5z^{6}-5z^{4}+3z^{2}}$ (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 \
-4-3-2-10123456χ
14          11
12         2 -2
10        51 4
8       73  -4
6      84   4
4     77    0
2    88     0
0   58      3
-2  37       -4
-4 15        4
-6 3         -3
-81          1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=0}$ ${\displaystyle i=2}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### 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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