# L10a167

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a167 at Knotilus!

### Link Presentations

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

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {-t(1)t(4)^{2}t(3)^{2}+t(1)t(2)t(4)^{2}t(3)^{2}-2t(2)t(4)^{2}t(3)^{2}+t(4)^{2}t(3)^{2}+t(1)t(4)t(3)^{2}+t(2)t(4)t(3)^{2}-t(4)t(3)^{2}+t(1)t(4)^{2}t(3)-t(1)t(2)t(4)^{2}t(3)+t(2)t(4)^{2}t(3)+t(1)t(3)+t(2)t(3)-2t(1)t(4)t(3)+t(1)t(2)t(4)t(3)-2t(2)t(4)t(3)+t(4)t(3)-t(3)-2t(1)+t(1)t(2)-t(2)+t(1)t(4)-t(1)t(2)t(4)+t(2)t(4)+1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}}$ (db) Jones polynomial ${\displaystyle 7q^{9/2}-10q^{7/2}+8q^{5/2}-{\frac {1}{q^{5/2}}}-9q^{3/2}+{\frac {1}{q^{3/2}}}-q^{15/2}+3q^{13/2}-6q^{11/2}+5{\sqrt {q}}-{\frac {5}{\sqrt {q}}}}$ (db) Signature 3 (db) HOMFLY-PT polynomial ${\displaystyle z^{7}a^{-3}-2z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}+az^{3}-9z^{3}a^{-1}+10z^{3}a^{-3}-3z^{3}a^{-5}+4az-14za^{-1}+13za^{-3}-3za^{-5}+4az^{-1}-11a^{-1}z^{-1}+10a^{-3}z^{-1}-3a^{-5}z^{-1}+az^{-3}-3a^{-1}z^{-3}+3a^{-3}z^{-3}-a^{-5}z^{-3}}$ (db) Kauffman polynomial ${\displaystyle -z^{9}a^{-1}-z^{9}a^{-3}-5z^{8}a^{-2}-4z^{8}a^{-4}-z^{8}-az^{7}-6z^{7}a^{-3}-7z^{7}a^{-5}+13z^{6}a^{-2}+4z^{6}a^{-4}-7z^{6}a^{-6}+2z^{6}+6az^{5}+14z^{5}a^{-1}+27z^{5}a^{-3}+13z^{5}a^{-5}-6z^{5}a^{-7}+6z^{4}a^{-2}+11z^{4}a^{-4}+8z^{4}a^{-6}-3z^{4}a^{-8}+6z^{4}-13az^{3}-30z^{3}a^{-1}-30z^{3}a^{-3}-6z^{3}a^{-5}+6z^{3}a^{-7}-z^{3}a^{-9}-33z^{2}a^{-2}-16z^{2}a^{-4}-17z^{2}+13az+28za^{-1}+21za^{-3}+3za^{-5}-3za^{-7}+24a^{-2}+11a^{-4}-a^{-6}+13-6az^{-1}-14a^{-1}z^{-1}-12a^{-3}z^{-1}-3a^{-5}z^{-1}+a^{-7}z^{-1}-6a^{-2}z^{-2}-3a^{-4}z^{-2}-3z^{-2}+az^{-3}+3a^{-1}z^{-3}+3a^{-3}z^{-3}+a^{-5}z^{-3}}$ (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 \
-4-3-2-10123456χ
16          11
14         31-2
12        3  3
10       43  -1
8      63   3
6     57    2
4    43     1
2   48      4
0  11       0
-2  4        4
-411         0
-61          1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=2}$ ${\displaystyle i=4}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

### 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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