# L10a111

Jump to navigationJump to search

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a111 at Knotilus!

### Link Presentations

 Planar diagram presentation X12,1,13,2 X16,7,17,8 X8394 X2,18,3,17 X14,6,15,5 X6,12,7,11 X18,10,19,9 X20,15,11,16 X10,13,1,14 X4,19,5,20 Gauss code {1, -4, 3, -10, 5, -6, 2, -3, 7, -9}, {6, -1, 9, -5, 8, -2, 4, -7, 10, -8}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}+t(1)^{2}-2t(2)^{2}t(1)+3t(2)t(1)-2t(1)+t(2)^{2}-2t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}}$ (db) Jones polynomial ${\displaystyle -{\frac {9}{q^{9/2}}}-q^{7/2}+{\frac {15}{q^{7/2}}}+5q^{5/2}-{\frac {19}{q^{5/2}}}-11q^{3/2}+{\frac {20}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+15{\sqrt {q}}-{\frac {20}{\sqrt {q}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle az^{7}-2a^{3}z^{5}+3az^{5}-z^{5}a^{-1}+a^{5}z^{3}-4a^{3}z^{3}+3az^{3}-z^{3}a^{-1}+a^{5}z-2a^{3}z+az+az^{-1}-a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -4a^{3}z^{9}-4az^{9}-9a^{4}z^{8}-20a^{2}z^{8}-11z^{8}-8a^{5}z^{7}-12a^{3}z^{7}-15az^{7}-11z^{7}a^{-1}-4a^{6}z^{6}+11a^{4}z^{6}+34a^{2}z^{6}-5z^{6}a^{-2}+14z^{6}-a^{7}z^{5}+12a^{5}z^{5}+34a^{3}z^{5}+39az^{5}+17z^{5}a^{-1}-z^{5}a^{-3}+5a^{6}z^{4}-3a^{4}z^{4}-14a^{2}z^{4}+4z^{4}a^{-2}-2z^{4}+a^{7}z^{3}-8a^{5}z^{3}-22a^{3}z^{3}-19az^{3}-6z^{3}a^{-1}-2a^{6}z^{2}-a^{4}z^{2}+2a^{2}z^{2}+z^{2}+2a^{5}z+4a^{3}z+2az+1-az^{-1}-a^{-1}z^{-1}}$ (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 \
-6-5-4-3-2-101234χ
8          11
6         4 -4
4        71 6
2       84  -4
0      127   5
-2     1010    0
-4    910     -1
-6   610      4
-8  39       -6
-10 16        5
-12 3         -3
-141          1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{12}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### 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).

Back to the top.