# L10a118

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a118 at Knotilus!
 Rich Schwartz' "25" [1] Two interlaced pentagons. Two hollow interlaced pentagrams.

### Link Presentations

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

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {-t(1)^{4}t(2)^{4}-t(1)^{3}t(2)^{3}-t(1)^{2}t(2)^{2}-t(1)t(2)-1}{t(1)^{2}t(2)^{2}}}}$ (db) Jones polynomial ${\displaystyle -{\frac {1}{q^{9/2}}}-{\frac {1}{q^{29/2}}}+{\frac {1}{q^{27/2}}}-{\frac {1}{q^{25/2}}}+{\frac {1}{q^{23/2}}}-{\frac {1}{q^{21/2}}}+{\frac {1}{q^{19/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{13/2}}}}$ (db) Signature -9 (db) HOMFLY-PT polynomial ${\displaystyle a^{11}z^{7}+7a^{11}z^{5}+15a^{11}z^{3}+10a^{11}z+a^{11}z^{-1}-a^{9}z^{9}-9a^{9}z^{7}-28a^{9}z^{5}-35a^{9}z^{3}-15a^{9}z-a^{9}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle a^{19}z+a^{18}z^{2}+a^{17}z^{3}-a^{17}z+a^{16}z^{4}-2a^{16}z^{2}+a^{15}z^{5}-3a^{15}z^{3}+a^{15}z+a^{14}z^{6}-4a^{14}z^{4}+3a^{14}z^{2}+a^{13}z^{7}-5a^{13}z^{5}+6a^{13}z^{3}-a^{13}z+a^{12}z^{8}-6a^{12}z^{6}+10a^{12}z^{4}-4a^{12}z^{2}+a^{11}z^{9}-8a^{11}z^{7}+22a^{11}z^{5}-25a^{11}z^{3}+11a^{11}z-a^{11}z^{-1}+a^{10}z^{8}-7a^{10}z^{6}+15a^{10}z^{4}-10a^{10}z^{2}+a^{10}+a^{9}z^{9}-9a^{9}z^{7}+28a^{9}z^{5}-35a^{9}z^{3}+15a^{9}z-a^{9}z^{-1}}$ (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 \
-10-9-8-7-6-5-4-3-2-10χ
-8          11
-10          11
-12        1  1
-14           0
-16      11   0
-18           0
-20    11     0
-22           0
-24  11       0
-26           0
-2811         0
-301          1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-10}$ ${\displaystyle i=-8}$ ${\displaystyle r=-10}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-9}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

### 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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