L10a121

L10a120

L10a122

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a121 at Knotilus!

Link Presentations

[edit Notes on L10a121's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{4}\left(-v^{2}\right)-3u^{3}v^{3}+5u^{3}v^{2}-3u^{3}v-u^{2}v^{4}+5u^{2}v^{3}-9u^{2}v^{2}+5u^{2}v-u^{2}-3uv^{3}+5uv^{2}-3uv-v^{2}}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {11}{q^{11/2}}}-{\frac {14}{q^{13/2}}}+{\frac {16}{q^{15/2}}}-{\frac {14}{q^{17/2}}}+{\frac {11}{q^{19/2}}}-{\frac {8}{q^{21/2}}}+{\frac {3}{q^{23/2}}}-{\frac {1}{q^{25/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{11}z^{3}+a^{11}z+a^{11}z^{-1}-a^{9}z^{5}+2a^{9}z-a^{9}z^{-1}-3a^{7}z^{5}-9a^{7}z^{3}-7a^{7}z-a^{5}z^{5}-2a^{5}z^{3}-a^{5}z$ (db)
Kauffman polynomial	$-z^{5}a^{15}+2z^{3}a^{15}-za^{15}-3z^{6}a^{14}+4z^{4}a^{14}-z^{2}a^{14}-6z^{7}a^{13}+12z^{5}a^{13}-12z^{3}a^{13}+7za^{13}-5z^{8}a^{12}+3z^{6}a^{12}+6z^{4}a^{12}-5z^{2}a^{12}-2z^{9}a^{11}-8z^{7}a^{11}+18z^{5}a^{11}-9z^{3}a^{11}+za^{11}+a^{11}z^{-1}-10z^{8}a^{10}+12z^{6}a^{10}+4z^{4}a^{10}-8z^{2}a^{10}-a^{10}-2z^{9}a^{9}-8z^{7}a^{9}+18z^{5}a^{9}-9z^{3}a^{9}+za^{9}+a^{9}z^{-1}-5z^{8}a^{8}+3z^{6}a^{8}+6z^{4}a^{8}-5z^{2}a^{8}-6z^{7}a^{7}+12z^{5}a^{7}-12z^{3}a^{7}+7za^{7}-3z^{6}a^{6}+4z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+2z^{3}a^{5}-za^{5}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                     1 1 -6                   3 1 -2 -8                 5     5 -10               6 3     -3 -12             8 5       3 -14           8 6         -2 -16         6 8           -2 -18       5 8             3 -20     3 6               -3 -22     5                 5 -24 1 3                   -2 -26 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-10$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{3}$ $r=-8$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-7$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-6$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

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