L10a60

L10a59

L10a61

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a60 at Knotilus!

Link Presentations

[edit Notes on L10a60's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X14,6,15,5 X18,13,19,14 X16,19,17,20 X12,17,13,18 X2738 X4,12,5,11 X6,16,1,15
Gauss code	{1, -8, 2, -9, 4, -10}, {8, -1, 3, -2, 9, -7, 5, -4, 10, -6, 7, -5, 6, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-3u^{2}v^{3}+4u^{2}v^{2}-2u^{2}v-uv^{4}+4uv^{3}-7uv^{2}+4uv-u-2v^{3}+4v^{2}-3v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{11/2}-3q^{9/2}+6q^{7/2}-10q^{5/2}+11q^{3/2}-13{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-2a^{3}z+3za^{-3}-a^{3}z^{-1}-z^{7}a^{-1}+2az^{5}-5z^{5}a^{-1}+7az^{3}-10z^{3}a^{-1}+8az-8za^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-z^{2}a^{-6}+3z^{5}a^{-5}-3z^{3}a^{-5}+za^{-5}+a^{4}z^{6}+5z^{6}a^{-4}-3a^{4}z^{4}-5z^{4}a^{-4}+3a^{4}z^{2}+2z^{2}a^{-4}-a^{4}+3a^{3}z^{7}+6z^{7}a^{-3}-9a^{3}z^{5}-8z^{5}a^{-3}+8a^{3}z^{3}+7z^{3}a^{-3}-3a^{3}z-3za^{-3}+a^{3}z^{-1}+3a^{2}z^{8}+4z^{8}a^{-2}-4a^{2}z^{6}-7a^{2}z^{4}-8z^{4}a^{-2}+10a^{2}z^{2}+6z^{2}a^{-2}-3a^{2}+az^{9}+z^{9}a^{-1}+8az^{7}+11z^{7}a^{-1}-29az^{5}-31z^{5}a^{-1}+28az^{3}+30z^{3}a^{-1}-13az-14za^{-1}+3az^{-1}+2a^{-1}z^{-1}+7z^{8}-10z^{6}-6z^{4}+10z^{2}-3$ (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 12                     1 -1 10                   2   2 8                 4 1   -3 6               6 2     4 4             6 5       -1 2           7 5         2 0         5 7           2 -2       4 6             -2 -4     2 5               3 -6   1 4                 -3 -8   2                   2 -10 1                     -1

Integral Khovanov Homology (db, data source)

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

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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