L11a522

L11a521

L11a523

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a522 at Knotilus!

Link Presentations

[edit Notes on L11a522's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X16,5,17,6 X18,8,19,7 X12,20,7,19 X10,14,11,13 X22,10,13,9 X20,15,21,16 X6,17,1,18 X2,11,3,12 X4,22,5,21
Gauss code	{1, -10, 2, -11, 3, -9}, {4, -1, 7, -6, 10, -5}, {6, -2, 8, -3, 9, -4, 5, -8, 11, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{2}w^{3}-2u^{2}v^{2}w^{2}+u^{2}v^{2}w-2u^{2}vw^{3}+4u^{2}vw^{2}-3u^{2}vw+u^{2}v+u^{2}w^{3}-2u^{2}w^{2}+u^{2}w-uv^{2}w^{3}+3uv^{2}w^{2}-4uv^{2}w+uv^{2}+3uvw^{3}-8uvw^{2}+8uvw-3uv-uw^{3}+4uw^{2}-3uw+u-v^{2}w^{2}+2v^{2}w-v^{2}-vw^{3}+3vw^{2}-4vw+2v-w^{2}+2w-1}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{6}-4q^{5}+10q^{4}-15q^{3}+22q^{2}-24q+25-21q^{-1}+16q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}+2z^{2}a^{-4}+a^{-4}z^{-2}+2a^{-4}-a^{2}z^{6}-2z^{6}a^{-2}-3a^{2}z^{4}-7z^{4}a^{-2}-3a^{2}z^{2}-9z^{2}a^{-2}-2a^{-2}z^{-2}-5a^{-2}+z^{8}+5z^{6}+10z^{4}+8z^{2}+z^{-2}+3$ (db)
Kauffman polynomial	$3z^{10}a^{-2}+3z^{10}+9az^{9}+17z^{9}a^{-1}+8z^{9}a^{-3}+11a^{2}z^{8}+13z^{8}a^{-2}+8z^{8}a^{-4}+16z^{8}+8a^{3}z^{7}-10az^{7}-36z^{7}a^{-1}-14z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-19a^{2}z^{6}-52z^{6}a^{-2}-19z^{6}a^{-4}+z^{6}a^{-6}-55z^{6}+a^{5}z^{5}-11a^{3}z^{5}-2az^{5}+21z^{5}a^{-1}+3z^{5}a^{-3}-8z^{5}a^{-5}-5a^{4}z^{4}+15a^{2}z^{4}+62z^{4}a^{-2}+16z^{4}a^{-4}-2z^{4}a^{-6}+64z^{4}-a^{5}z^{3}+4a^{3}z^{3}+9az^{3}+5z^{3}a^{-1}+4z^{3}a^{-3}+3z^{3}a^{-5}+a^{4}z^{2}-7a^{2}z^{2}-37z^{2}a^{-2}-12z^{2}a^{-4}+z^{2}a^{-6}-32z^{2}-a^{3}z-3az-8za^{-1}-6za^{-3}+a^{2}+12a^{-2}+6a^{-4}+8+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   7 1   6 7                 9 4     -5 5               13 6       7 3             12 10         -2 1           13 12           1 -1         9 13             4 -3       7 12               -5 -5     3 10                 7 -7   1 6                   -5 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

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

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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