L11a526

L11a525

L11a527

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a526 at Knotilus!

Link Presentations

[edit Notes on L11a526's Link Presentations]

Planar diagram presentation	X8192 X14,3,15,4 X12,15,7,16 X10,22,11,21 X16,9,17,10 X20,12,21,11 X2738 X22,17,13,18 X6,13,1,14 X4,20,5,19 X18,6,19,5
Gauss code	{1, -7, 2, -10, 11, -9}, {7, -1, 5, -4, 6, -3}, {9, -2, 3, -5, 8, -11, 10, -6, 4, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(3)-1)\left(-t(2)^{2}t(1)^{2}+t(2)t(3)^{2}t(1)^{2}-t(3)^{2}t(1)^{2}+2t(2)t(1)^{2}-2t(2)t(3)t(1)^{2}+t(3)t(1)^{2}-t(1)^{2}+2t(2)^{2}t(1)+t(2)^{2}t(3)^{2}t(1)-3t(2)t(3)^{2}t(1)+2t(3)^{2}t(1)-3t(2)t(1)-2t(2)^{2}t(3)t(1)+6t(2)t(3)t(1)-2t(3)t(1)+t(1)-t(2)^{2}-t(2)^{2}t(3)^{2}+2t(2)t(3)^{2}-t(3)^{2}+t(2)+t(2)^{2}t(3)-2t(2)t(3)\right)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{3}-4q^{2}+10q-16+23q^{-1}-25q^{-2}+27q^{-3}-22q^{-4}+17q^{-5}-10q^{-6}+4q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{4}\right)-a^{6}z^{2}-2a^{6}+a^{4}z^{6}+2a^{4}z^{4}+6a^{4}z^{2}+a^{4}z^{-2}+6a^{4}+a^{2}z^{6}-a^{2}z^{4}-6a^{2}z^{2}-2a^{2}z^{-2}+z^{2}a^{-2}-7a^{2}-2z^{4}+z^{-2}+3$ (db)
Kauffman polynomial	$a^{9}z^{5}-a^{9}z^{3}+4a^{8}z^{6}-4a^{8}z^{4}+9a^{7}z^{7}-13a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z+13a^{6}z^{8}-25a^{6}z^{6}+24a^{6}z^{4}-13a^{6}z^{2}+4a^{6}+10a^{5}z^{9}-10a^{5}z^{7}-5a^{5}z^{5}+13a^{5}z^{3}-6a^{5}z+3a^{4}z^{10}+20a^{4}z^{8}-64a^{4}z^{6}+73a^{4}z^{4}-42a^{4}z^{2}-a^{4}z^{-2}+13a^{4}+18a^{3}z^{9}-35a^{3}z^{7}+18a^{3}z^{5}+3a^{3}z^{3}-7a^{3}z+2a^{3}z^{-1}+3a^{2}z^{10}+15a^{2}z^{8}-54a^{2}z^{6}+z^{6}a^{-2}+62a^{2}z^{4}-2z^{4}a^{-2}-40a^{2}z^{2}+z^{2}a^{-2}-2a^{2}z^{-2}+13a^{2}+8az^{9}-12az^{7}+4z^{7}a^{-1}+az^{5}-8z^{5}a^{-1}+2az^{3}+4z^{3}a^{-1}-3az+2az^{-1}+8z^{8}-18z^{6}+15z^{4}-10z^{2}-z^{-2}+5$ (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 4 χ 7                       1 1 5                     3   -3 3                   7 1   6 1                 9 3     -6 -1               14 7       7 -3             13 11         -2 -5           14 12           2 -7         10 15             5 -9       7 12               -5 -11     3 10                 7 -13   1 7                   -6 -15   3                     3 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{14}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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