# L11a313

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a313 at Knotilus!

### Link Presentations

 Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,6,19,5 X14,8,15,7 X20,13,21,14 X8,21,1,22 X16,10,17,9 X22,16,9,15 X6,18,7,17 X4,20,5,19 Gauss code {1, -2, 3, -11, 4, -10, 5, -7}, {8, -1, 2, -3, 6, -5, 9, -8, 10, -4, 11, -6, 7, -9}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(t(1)-1)(t(2)-1)(t(1)+t(2))(t(1)t(2)+1)\left(t(2)^{2}-t(2)+1\right)}{t(1)^{3/2}t(2)^{5/2}}}}$ (db) Jones polynomial ${\displaystyle -q^{11/2}+3q^{9/2}-6q^{7/2}+10q^{5/2}-13q^{3/2}+14{\sqrt {q}}-{\frac {16}{\sqrt {q}}}+{\frac {13}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{11/2}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle -a^{3}z^{5}-z^{5}a^{-3}-3a^{3}z^{3}-3z^{3}a^{-3}-2a^{3}z-2za^{-3}+az^{7}+z^{7}a^{-1}+4az^{5}+4z^{5}a^{-1}+5az^{3}+5z^{3}a^{-1}+2az+2za^{-1}+az^{-1}-a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -2z^{10}a^{-2}-2z^{10}-5az^{9}-9z^{9}a^{-1}-4z^{9}a^{-3}-7a^{2}z^{8}+z^{8}a^{-2}-3z^{8}a^{-4}-3z^{8}-7a^{3}z^{7}+7az^{7}+30z^{7}a^{-1}+15z^{7}a^{-3}-z^{7}a^{-5}-5a^{4}z^{6}+13a^{2}z^{6}+13z^{6}a^{-2}+12z^{6}a^{-4}+19z^{6}-3a^{5}z^{5}+12a^{3}z^{5}+4az^{5}-33z^{5}a^{-1}-18z^{5}a^{-3}+4z^{5}a^{-5}-a^{6}z^{4}+5a^{4}z^{4}-9a^{2}z^{4}-16z^{4}a^{-2}-14z^{4}a^{-4}-17z^{4}+3a^{5}z^{3}-11a^{3}z^{3}-11az^{3}+19z^{3}a^{-1}+12z^{3}a^{-3}-4z^{3}a^{-5}+a^{6}z^{2}-a^{4}z^{2}+7z^{2}a^{-2}+5z^{2}a^{-4}+4z^{2}+4a^{3}z+4az-4za^{-1}-4za^{-3}+1-az^{-1}-a^{-1}z^{-1}}$ (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       74   3
2      76    -1
0     97     2
-2    69      3
-4   47       -3
-6  26        4
-8 14         -3
-10 2          2
-121           -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### 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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