# L11a92

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a92 at Knotilus!

### Link Presentations

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

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {t(2)^{5}+3t(1)t(2)^{4}-6t(2)^{4}-10t(1)t(2)^{3}+11t(2)^{3}+11t(1)t(2)^{2}-10t(2)^{2}-6t(1)t(2)+3t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}}$ (db) Jones polynomial ${\displaystyle -{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {13}{q^{5/2}}}-{\frac {18}{q^{7/2}}}+{\frac {20}{q^{9/2}}}-{\frac {20}{q^{11/2}}}+{\frac {17}{q^{13/2}}}-{\frac {13}{q^{15/2}}}+{\frac {7}{q^{17/2}}}-{\frac {3}{q^{19/2}}}+{\frac {1}{q^{21/2}}}}$ (db) Signature -3 (db) HOMFLY-PT polynomial ${\displaystyle -a^{11}z^{-1}+4za^{9}+4a^{9}z^{-1}-6z^{3}a^{7}-11za^{7}-6a^{7}z^{-1}+3z^{5}a^{5}+8z^{3}a^{5}+10za^{5}+5a^{5}z^{-1}+z^{5}a^{3}-z^{3}a^{3}-4za^{3}-2a^{3}z^{-1}-z^{3}a-za}$ (db) Kauffman polynomial ${\displaystyle a^{12}z^{6}-3a^{12}z^{4}+3a^{12}z^{2}-a^{12}+3a^{11}z^{7}-8a^{11}z^{5}+8a^{11}z^{3}-4a^{11}z+a^{11}z^{-1}+4a^{10}z^{8}-5a^{10}z^{6}-5a^{10}z^{4}+9a^{10}z^{2}-3a^{10}+3a^{9}z^{9}+5a^{9}z^{7}-27a^{9}z^{5}+31a^{9}z^{3}-18a^{9}z+4a^{9}z^{-1}+a^{8}z^{10}+11a^{8}z^{8}-22a^{8}z^{6}+4a^{8}z^{4}+7a^{8}z^{2}-3a^{8}+7a^{7}z^{9}+4a^{7}z^{7}-41a^{7}z^{5}+52a^{7}z^{3}-29a^{7}z+6a^{7}z^{-1}+a^{6}z^{10}+14a^{6}z^{8}-28a^{6}z^{6}+15a^{6}z^{4}-a^{6}+4a^{5}z^{9}+8a^{5}z^{7}-33a^{5}z^{5}+41a^{5}z^{3}-23a^{5}z+5a^{5}z^{-1}+7a^{4}z^{8}-9a^{4}z^{6}+5a^{4}z^{4}-a^{4}+6a^{3}z^{7}-10a^{3}z^{5}+10a^{3}z^{3}-7a^{3}z+2a^{3}z^{-1}+3a^{2}z^{6}-4a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az}$ (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 \
-9-8-7-6-5-4-3-2-1012χ
2           11
0          2 -2
-2         61 5
-4        83  -5
-6       105   5
-8      108    -2
-10     1010     0
-12    811      3
-14   59       -4
-16  28        6
-18 15         -4
-20 2          2
-221           -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-4}$ ${\displaystyle i=-2}$ ${\displaystyle r=-9}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### 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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