# L11a47

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a47 at Knotilus!

### Link Presentations

 Planar diagram presentation X6172 X16,7,17,8 X18,9,19,10 X8,17,9,18 X4,19,1,20 X12,6,13,5 X10,4,11,3 X22,14,5,13 X14,22,15,21 X20,12,21,11 X2,16,3,15 Gauss code {1, -11, 7, -5}, {6, -1, 2, -4, 3, -7, 10, -6, 8, -9, 11, -2, 4, -3, 5, -10, 9, -8}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(u-1)(v-1)\left(v^{2}+1\right)\left(2v^{2}-3v+2\right)}{{\sqrt {u}}v^{5/2}}}}$ (db) Jones polynomial ${\displaystyle 3q^{9/2}-{\frac {4}{q^{9/2}}}-6q^{7/2}+{\frac {8}{q^{7/2}}}+10q^{5/2}-{\frac {13}{q^{5/2}}}-15q^{3/2}+{\frac {16}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+17{\sqrt {q}}-{\frac {18}{\sqrt {q}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle -a^{3}z^{5}-z^{5}a^{-3}-2a^{3}z^{3}-3z^{3}a^{-3}-2za^{-3}+a^{3}z^{-1}-a^{-3}z^{-1}+az^{7}+z^{7}a^{-1}+3az^{5}+4z^{5}a^{-1}+az^{3}+6z^{3}a^{-1}-3az+5za^{-1}-2az^{-1}+2a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle a^{6}z^{4}+z^{7}a^{-5}+4a^{5}z^{5}-4z^{5}a^{-5}-2a^{5}z^{3}+4z^{3}a^{-5}+3z^{8}a^{-4}+8a^{4}z^{6}-12z^{6}a^{-4}-7a^{4}z^{4}+14z^{4}a^{-4}+a^{4}z^{2}-4z^{2}a^{-4}+4z^{9}a^{-3}+11a^{3}z^{7}-15z^{7}a^{-3}-15a^{3}z^{5}+20z^{5}a^{-3}+6a^{3}z^{3}-14z^{3}a^{-3}+a^{3}z+5za^{-3}-a^{3}z^{-1}-a^{-3}z^{-1}+2z^{10}a^{-2}+10a^{2}z^{8}-14a^{2}z^{6}-14z^{6}a^{-2}+4a^{2}z^{4}+16z^{4}a^{-2}+a^{2}z^{2}-5z^{2}a^{-2}+6az^{9}+10z^{9}a^{-1}-4az^{7}-31z^{7}a^{-1}-6az^{5}+37z^{5}a^{-1}-3az^{3}-29z^{3}a^{-1}+8az+12za^{-1}-2az^{-1}-2a^{-1}z^{-1}+2z^{10}+7z^{8}-24z^{6}+14z^{4}-z^{2}-1}$ (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 \
-5-4-3-2-10123456χ
12           11
10          2 -2
8         41 3
6        62  -4
4       94   5
2      86    -2
0     109     1
-2    810      2
-4   58       -3
-6  38        5
-8 15         -4
-10 3          3
-121           -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### 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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