L10a129

L10a128

L10a130

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a129 at Knotilus!

Link Presentations

[edit Notes on L10a129's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X16,9,17,10 X14,8,15,7 X18,15,19,16 X20,14,11,13 X10,17,5,18 X8,20,9,19 X2536 X4,11,1,12
Gauss code	{1, -9, 2, -10}, {9, -1, 4, -8, 3, -7}, {10, -2, 6, -4, 5, -3, 7, -5, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-2t(1)t(3)t(2)^{2}+3t(3)t(2)^{2}-t(2)^{2}-2t(1)t(3)^{2}t(2)+3t(3)^{2}t(2)-3t(1)t(2)+6t(1)t(3)t(2)-6t(3)t(2)+2t(2)+t(1)t(3)^{2}-t(3)^{2}+2t(1)-3t(1)t(3)+2t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{2}+4q-7+11q^{-1}-13q^{-2}+15q^{-3}-12q^{-4}+11q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{8}+3z^{2}a^{6}+a^{6}z^{-2}+3a^{6}-3z^{4}a^{4}-6z^{2}a^{4}-2a^{4}z^{-2}-5a^{4}+z^{6}a^{2}+3z^{4}a^{2}+5z^{2}a^{2}+a^{2}z^{-2}+3a^{2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-6z^{4}a^{8}+5z^{2}a^{8}-2a^{8}+4z^{7}a^{7}-4z^{5}a^{7}-2z^{3}a^{7}+3za^{7}+3z^{8}a^{6}+4z^{6}a^{6}-17z^{4}a^{6}+17z^{2}a^{6}+a^{6}z^{-2}-8a^{6}+z^{9}a^{5}+10z^{7}a^{5}-18z^{5}a^{5}+5z^{3}a^{5}+5za^{5}-2a^{5}z^{-1}+7z^{8}a^{4}-z^{6}a^{4}-19z^{4}a^{4}+20z^{2}a^{4}+2a^{4}z^{-2}-9a^{4}+z^{9}a^{3}+12z^{7}a^{3}-24z^{5}a^{3}+10z^{3}a^{3}+3za^{3}-2a^{3}z^{-1}+4z^{8}a^{2}+2z^{6}a^{2}-15z^{4}a^{2}+11z^{2}a^{2}+a^{2}z^{-2}-4a^{2}+6z^{7}a-10z^{5}a+4z^{3}a+4z^{6}-7z^{4}+3z^{2}+z^{5}a^{-1}-z^{3}a^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                     1 -1 3                   3   3 1                 4 1   -3 -1               7 3     4 -3             7 5       -2 -5           8 6         2 -7         6 9           3 -9       5 6             -1 -11     2 7               5 -13   1 4                 -3 -15   2                   2 -17 1                     -1

Integral Khovanov Homology (db, data source)

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

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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