L10a132

L10a131

L10a133

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a132 at Knotilus!

Link Presentations

[edit Notes on L10a132's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}-2t(1)t(3)t(2)^{2}+2t(3)t(2)^{2}-2t(1)t(3)^{2}t(2)+2t(3)^{2}t(2)-2t(1)t(2)+4t(1)t(3)t(2)-4t(3)t(2)+2t(2)+2t(1)-2t(1)t(3)+2t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{-10}-2q^{-9}+5q^{-8}-7q^{-7}+10q^{-6}-9q^{-5}+10q^{-4}-7q^{-3}+5q^{-2}-3q^{-1}+1$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}+a^{10}-3a^{8}z^{2}-2a^{8}z^{-2}-6a^{8}+3a^{6}z^{4}+8a^{6}z^{2}+a^{6}z^{-2}+5a^{6}-a^{4}z^{6}-3a^{4}z^{4}-2a^{4}z^{2}+a^{2}z^{4}+2a^{2}z^{2}$ (db)
Kauffman polynomial	$z^{4}a^{12}-2z^{2}a^{12}+a^{12}+2z^{5}a^{11}-2z^{3}a^{11}+3z^{6}a^{10}-3z^{4}a^{10}+3z^{2}a^{10}+a^{10}z^{-2}-3a^{10}+3z^{7}a^{9}-z^{5}a^{9}-3z^{3}a^{9}+6za^{9}-2a^{9}z^{-1}+2z^{8}a^{8}+3z^{6}a^{8}-12z^{4}a^{8}+15z^{2}a^{8}+2a^{8}z^{-2}-8a^{8}+z^{9}a^{7}+3z^{7}a^{7}-5z^{5}a^{7}-3z^{3}a^{7}+6za^{7}-2a^{7}z^{-1}+5z^{8}a^{6}-9z^{6}a^{6}+6z^{2}a^{6}+a^{6}z^{-2}-5a^{6}+z^{9}a^{5}+3z^{7}a^{5}-12z^{5}a^{5}+6z^{3}a^{5}+3z^{8}a^{4}-8z^{6}a^{4}+5z^{4}a^{4}-2z^{2}a^{4}+3z^{7}a^{3}-10z^{5}a^{3}+8z^{3}a^{3}+z^{6}a^{2}-3z^{4}a^{2}+2z^{2}a^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                     1 1 -1                   2   -2 -3                 3 1   2 -5               5 3     -2 -7             5 2       3 -9           4 5         1 -11         6 5           1 -13       2 5             3 -15     3 5               -2 -17   1 4                 3 -19   1                   -1 -21 1                     1

Integral Khovanov Homology (db, data source)

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

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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