# L11a453

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a453 at Knotilus!

### Link Presentations

 Planar diagram presentation X6172 X14,4,15,3 X18,11,19,12 X16,8,17,7 X20,17,21,18 X12,19,5,20 X8,22,9,21 X22,10,13,9 X10,14,11,13 X2536 X4,16,1,15 Gauss code {1, -10, 2, -11}, {10, -1, 4, -7, 8, -9, 3, -6}, {9, -2, 11, -4, 5, -3, 6, -5, 7, -8}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {(vw-v+1)(vw-w+1)(uvw-uw+u-vw+v-1)}{{\sqrt {u}}v^{3/2}w^{3/2}}}}$ (db) Jones polynomial ${\displaystyle q^{7}-3q^{6}+7q^{5}-12q^{4}+16q^{3}-16q^{2}+18q-14+11q^{-1}-6q^{-2}+3q^{-3}-q^{-4}}$ (db) Signature 2 (db) HOMFLY-PT polynomial ${\displaystyle z^{6}a^{-4}+4z^{4}a^{-4}+6z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-z^{8}a^{-2}-6z^{6}a^{-2}-a^{2}z^{4}-15z^{4}a^{-2}-3a^{2}z^{2}-18z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}-9a^{-2}+2z^{6}+9z^{4}+14z^{2}+z^{-2}+8}$ (db) Kauffman polynomial ${\displaystyle z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-2z^{3}a^{-7}+6z^{6}a^{-6}-6z^{4}a^{-6}+4z^{2}a^{-6}-a^{-6}+9z^{7}a^{-5}-14z^{5}a^{-5}+11z^{3}a^{-5}-za^{-5}+9z^{8}a^{-4}-15z^{6}a^{-4}+11z^{4}a^{-4}-6z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+5z^{9}a^{-3}+a^{3}z^{7}+z^{7}a^{-3}-4a^{3}z^{5}-22z^{5}a^{-3}+5a^{3}z^{3}+22z^{3}a^{-3}-2a^{3}z-8za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+3a^{2}z^{8}+15z^{8}a^{-2}-12a^{2}z^{6}-51z^{6}a^{-2}+16a^{2}z^{4}+58z^{4}a^{-2}-9a^{2}z^{2}-37z^{2}a^{-2}-2a^{-2}z^{-2}+3a^{2}+11a^{-2}+3az^{9}+8z^{9}a^{-1}-6az^{7}-15z^{7}a^{-1}-6az^{5}-7z^{5}a^{-1}+16az^{3}+20z^{3}a^{-1}-7az-12za^{-1}+2a^{-1}z^{-1}+z^{10}+9z^{8}-42z^{6}+56z^{4}-35z^{2}-z^{-2}+11}$ (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 \
-5-4-3-2-10123456χ
15           11
13          2 -2
11         51 4
9        83  -5
7       84   4
5      88    0
3     108     2
1    711      4
-1   47       -3
-3  27        5
-5 14         -3
-7 2          2
-91           -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### 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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