L10a131

L10a130

L10a132

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a131 at Knotilus!

Link Presentations

[edit Notes on L10a131's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}-t(1)t(3)t(2)^{2}+3t(3)t(2)^{2}-t(2)^{2}-3t(1)t(3)^{2}t(2)+t(3)^{2}t(2)-t(1)t(2)+4t(1)t(3)t(2)-4t(3)t(2)+3t(2)+t(1)t(3)^{2}+2t(1)-3t(1)t(3)+t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{10}-3q^{9}+6q^{8}-9q^{7}+11q^{6}-11q^{5}+11q^{4}-7q^{3}+6q^{2}-2q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+3z^{2}a^{-2}-3z^{2}a^{-4}-3z^{2}a^{-6}+2z^{2}a^{-8}+3a^{-2}-3a^{-4}-a^{-6}+a^{-8}+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$z^{9}a^{-5}+z^{9}a^{-7}+2z^{8}a^{-4}+6z^{8}a^{-6}+4z^{8}a^{-8}+2z^{7}a^{-3}+4z^{7}a^{-5}+8z^{7}a^{-7}+6z^{7}a^{-9}+z^{6}a^{-2}-z^{6}a^{-4}-10z^{6}a^{-6}-3z^{6}a^{-8}+5z^{6}a^{-10}-5z^{5}a^{-3}-14z^{5}a^{-5}-22z^{5}a^{-7}-10z^{5}a^{-9}+3z^{5}a^{-11}-4z^{4}a^{-2}-10z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}a^{-8}-6z^{4}a^{-10}+z^{4}a^{-12}+z^{3}a^{-3}+8z^{3}a^{-5}+19z^{3}a^{-7}+9z^{3}a^{-9}-3z^{3}a^{-11}+6z^{2}a^{-2}+12z^{2}a^{-4}+6z^{2}a^{-6}+4z^{2}a^{-8}+3z^{2}a^{-10}-z^{2}a^{-12}+4za^{-3}-6za^{-7}-2za^{-9}-4a^{-2}-6a^{-4}-3a^{-6}-a^{-8}-a^{-10}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 21                     1 1 19                   2   -2 17                 4 1   3 15               6 3     -3 13             5 3       2 11           6 6         0 9         5 5           0 7       3 7             4 5     3 4               -1 3   1 5                 4 1   1                   -1 -1 1                     1

Integral Khovanov Homology (db, data source)

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

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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