# L11a395

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a395 at Knotilus!

### Link Presentations

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

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {uvw^{3}-4uvw^{2}+4uvw-2uv-2uw^{3}+5uw^{2}-5uw+2u-2vw^{3}+5vw^{2}-5vw+2v+2w^{3}-4w^{2}+4w-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}}$ (db) Jones polynomial ${\displaystyle 1-4q^{-1}+8q^{-2}-11q^{-3}+15q^{-4}-15q^{-5}+17q^{-6}-12q^{-7}+9q^{-8}-5q^{-9}+2q^{-10}-q^{-11}}$ (db) Signature -4 (db) HOMFLY-PT polynomial ${\displaystyle -a^{12}z^{-2}+4a^{10}z^{-2}+4a^{10}-6z^{2}a^{8}-5a^{8}z^{-2}-11a^{8}+4z^{4}a^{6}+9z^{2}a^{6}+2a^{6}z^{-2}+7a^{6}-z^{6}a^{4}-2z^{4}a^{4}-z^{2}a^{4}+z^{4}a^{2}+z^{2}a^{2}}$ (db) Kauffman polynomial ${\displaystyle z^{5}a^{13}-3z^{3}a^{13}+3za^{13}-a^{13}z^{-1}+2z^{6}a^{12}-4z^{4}a^{12}+3z^{2}a^{12}+a^{12}z^{-2}-2a^{12}+2z^{7}a^{11}+2z^{5}a^{11}-12z^{3}a^{11}+13za^{11}-5a^{11}z^{-1}+2z^{8}a^{10}+4z^{6}a^{10}-13z^{4}a^{10}+14z^{2}a^{10}+4a^{10}z^{-2}-10a^{10}+2z^{9}a^{9}+z^{7}a^{9}+3z^{5}a^{9}-16z^{3}a^{9}+21za^{9}-9a^{9}z^{-1}+z^{10}a^{8}+4z^{8}a^{8}-4z^{6}a^{8}-8z^{4}a^{8}+20z^{2}a^{8}+5a^{8}z^{-2}-14a^{8}+6z^{9}a^{7}-9z^{7}a^{7}+4z^{5}a^{7}-6z^{3}a^{7}+11za^{7}-5a^{7}z^{-1}+z^{10}a^{6}+8z^{8}a^{6}-22z^{6}a^{6}+11z^{4}a^{6}+6z^{2}a^{6}+2a^{6}z^{-2}-7a^{6}+4z^{9}a^{5}-4z^{7}a^{5}-8z^{5}a^{5}+6z^{3}a^{5}+6z^{8}a^{4}-15z^{6}a^{4}+8z^{4}a^{4}-2z^{2}a^{4}+4z^{7}a^{3}-10z^{5}a^{3}+5z^{3}a^{3}+z^{6}a^{2}-2z^{4}a^{2}+z^{2}a^{2}}$ (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 \
-9-8-7-6-5-4-3-2-1012χ
1           11
-1          3 -3
-3         51 4
-5        74  -3
-7       84   4
-9      77    0
-11     108     2
-13    510      5
-15   47       -3
-17  15        4
-19 14         -3
-21 1          1
-231           -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-5}$ ${\displaystyle i=-3}$ ${\displaystyle r=-9}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### 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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