L10a103

L10a102

L10a104

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a103 at Knotilus!

Link Presentations

[edit Notes on L10a103's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(2)t(1)-t(1)+1)(t(1)t(2)-t(2)+1)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {1}{q^{9/2}}}+6q^{7/2}-{\frac {3}{q^{7/2}}}-10q^{5/2}+{\frac {5}{q^{5/2}}}+11q^{3/2}-{\frac {9}{q^{3/2}}}+q^{11/2}-12{\sqrt {q}}+{\frac {11}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-5z^{5}a^{-1}+z^{5}a^{-3}-a^{3}z^{3}+7az^{3}-10z^{3}a^{-1}+3z^{3}a^{-3}-2a^{3}z+7az-8za^{-1}+3za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-3a^{2}z^{8}-4z^{8}a^{-2}-7z^{8}-3a^{3}z^{7}-7az^{7}-10z^{7}a^{-1}-6z^{7}a^{-3}-a^{4}z^{6}+6a^{2}z^{6}+z^{6}a^{-2}-5z^{6}a^{-4}+13z^{6}+10a^{3}z^{5}+27az^{5}+28z^{5}a^{-1}+8z^{5}a^{-3}-3z^{5}a^{-5}+3a^{4}z^{4}+5z^{4}a^{-2}+5z^{4}a^{-4}-z^{4}a^{-6}-4z^{4}-10a^{3}z^{3}-27az^{3}-26z^{3}a^{-1}-6z^{3}a^{-3}+3z^{3}a^{-5}-2a^{4}z^{2}-2a^{2}z^{2}-3z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}a^{-6}+3a^{3}z+10az+10za^{-1}+2za^{-3}-za^{-5}+1-az^{-1}-a^{-1}z^{-1}$ (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 12                     1 -1 10                   2   2 8                 4 1   -3 6               6 2     4 4             5 4       -1 2           7 6         1 0         6 7           1 -2       3 5             -2 -4     2 6               4 -6   1 3                 -2 -8   2                   2 -10 1                     -1

Integral Khovanov Homology (db, data source)

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

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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