L10a112

L10a111

L10a113

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a112 at Knotilus!

Link Presentations

[edit Notes on L10a112's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X14,6,15,5 X18,7,19,8 X16,10,17,9 X10,11,1,12 X6,14,7,13 X4,19,5,20 X20,15,11,16 X2,18,3,17
Gauss code	{1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, {6, -1, 7, -3, 9, -5, 10, -4, 8, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(u^{2}v^{2}-u^{2}v+u^{2}-2uv^{2}+3uv-2u+v^{2}-v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {4}{q^{9/2}}}-4q^{7/2}+{\frac {9}{q^{7/2}}}+8q^{5/2}-{\frac {14}{q^{5/2}}}-13q^{3/2}+{\frac {16}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+16{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-a^{3}z^{5}+4az^{5}-2z^{5}a^{-1}-2a^{3}z^{3}+7az^{3}-5z^{3}a^{-1}+z^{3}a^{-3}-2a^{3}z+5az-4za^{-1}+za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}+4a^{5}z^{5}-a^{5}z^{3}+9a^{4}z^{6}+z^{6}a^{-4}-8a^{4}z^{4}-2z^{4}a^{-4}+3a^{4}z^{2}+z^{2}a^{-4}+13a^{3}z^{7}+4z^{7}a^{-3}-20a^{3}z^{5}-10z^{5}a^{-3}+13a^{3}z^{3}+8z^{3}a^{-3}-4a^{3}z-2za^{-3}+10a^{2}z^{8}+6z^{8}a^{-2}-10a^{2}z^{6}-13z^{6}a^{-2}-3a^{2}z^{4}+6z^{4}a^{-2}+3a^{2}z^{2}+z^{2}a^{-2}+3az^{9}+3z^{9}a^{-1}+16az^{7}+7z^{7}a^{-1}-47az^{5}-33z^{5}a^{-1}+34az^{3}+28z^{3}a^{-1}-10az-8za^{-1}+az^{-1}+a^{-1}z^{-1}+16z^{8}-33z^{6}+14z^{4}-1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                     1 -1 8                   3   3 6                 5 1   -4 4               8 3     5 2             8 5       -3 0           10 8         2 -2         8 10           2 -4       6 8             -2 -6     3 8               5 -8   1 6                 -5 -10   3                   3 -12 1                     -1

Integral Khovanov Homology (db, data source)

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

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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