L11a507

L11a506

L11a508

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a507 at Knotilus!

Link Presentations

[edit Notes on L11a507's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(w-1)\left(u^{2}v^{2}w+u^{2}\left(-v^{2}\right)+u^{2}vw^{2}-2u^{2}vw+u^{2}v-u^{2}w^{2}+u^{2}w-2uv^{2}w+2uv^{2}-2uvw^{2}+5uvw-2uv+2uw^{2}-2uw+v^{2}w-v^{2}+vw^{2}-2vw+v-w^{2}+w\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{9}-4q^{8}+8q^{7}-13q^{6}+19q^{5}-20q^{4}+22q^{3}-18q^{2}-q^{-2}+14q+4q^{-1}-8$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-8}-2z^{4}a^{-6}-2z^{2}a^{-6}+a^{-6}z^{-2}+z^{6}a^{-4}+z^{4}a^{-4}-2a^{-4}z^{-2}-2a^{-4}+z^{6}a^{-2}+2z^{4}a^{-2}+3z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$2z^{10}a^{-4}+2z^{10}a^{-6}+6z^{9}a^{-3}+12z^{9}a^{-5}+6z^{9}a^{-7}+8z^{8}a^{-2}+12z^{8}a^{-4}+11z^{8}a^{-6}+7z^{8}a^{-8}+7z^{7}a^{-1}-3z^{7}a^{-3}-24z^{7}a^{-5}-10z^{7}a^{-7}+4z^{7}a^{-9}-10z^{6}a^{-2}-35z^{6}a^{-4}-41z^{6}a^{-6}-19z^{6}a^{-8}+z^{6}a^{-10}+4z^{6}+az^{5}-10z^{5}a^{-1}-6z^{5}a^{-3}+14z^{5}a^{-5}-z^{5}a^{-7}-10z^{5}a^{-9}+2z^{4}a^{-2}+34z^{4}a^{-4}+44z^{4}a^{-6}+16z^{4}a^{-8}-2z^{4}a^{-10}-6z^{4}-az^{3}+3z^{3}a^{-1}+4z^{3}a^{-3}+5z^{3}a^{-7}+5z^{3}a^{-9}+z^{2}a^{-2}-10z^{2}a^{-4}-15z^{2}a^{-6}-6z^{2}a^{-8}+2z^{2}+2za^{-3}+2za^{-5}-2a^{-2}-3a^{-4}-2a^{-6}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3   -3 15                   5 1   4 13                 8 3     -5 11               11 5       6 9             10 9         -1 7           12 10           2 5         8 12             4 3       6 10               -4 1     3 9                 6 -1   1 5                   -4 -3   3                     3 -5 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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