L11a505

L11a504

L11a506

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a505 at Knotilus!

Link Presentations

[edit Notes on L11a505's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(w-1)\left(u^{2}v^{2}w^{2}-2u^{2}v^{2}w-u^{2}vw^{2}+3u^{2}vw-u^{2}v-u^{2}w+u^{2}-2uv^{2}w^{2}+3uv^{2}w+2uvw^{2}-7uvw+2uv+3uw-2u+v^{2}w^{2}-v^{2}w-vw^{2}+3vw-v-2w+1\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-9q^{6}+17q^{5}-22q^{4}+27q^{3}-26q^{2}+24q-17+11q^{-1}-5q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-2z^{2}a^{-6}+a^{-6}z^{-2}+2z^{6}a^{-4}+6z^{4}a^{-4}+4z^{2}a^{-4}-2a^{-4}z^{-2}-2a^{-4}-z^{8}a^{-2}-4z^{6}a^{-2}-5z^{4}a^{-2}-z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}+z^{6}+2z^{4}$ (db)
Kauffman polynomial	$3z^{10}a^{-2}+3z^{10}a^{-4}+9z^{9}a^{-1}+18z^{9}a^{-3}+9z^{9}a^{-5}+18z^{8}a^{-2}+19z^{8}a^{-4}+11z^{8}a^{-6}+10z^{8}+5az^{7}-11z^{7}a^{-1}-30z^{7}a^{-3}-6z^{7}a^{-5}+8z^{7}a^{-7}+a^{2}z^{6}-57z^{6}a^{-2}-54z^{6}a^{-4}-16z^{6}a^{-6}+4z^{6}a^{-8}-22z^{6}-9az^{5}-8z^{5}a^{-1}+6z^{5}a^{-3}-6z^{5}a^{-5}-10z^{5}a^{-7}+z^{5}a^{-9}-a^{2}z^{4}+46z^{4}a^{-2}+48z^{4}a^{-4}+11z^{4}a^{-6}-5z^{4}a^{-8}+13z^{4}+3az^{3}+8z^{3}a^{-1}+5z^{3}a^{-3}+5z^{3}a^{-5}+4z^{3}a^{-7}-z^{3}a^{-9}-11z^{2}a^{-2}-14z^{2}a^{-4}-3z^{2}a^{-6}+2z^{2}a^{-8}-2z^{2}+2za^{-3}+2za^{-5}-2a^{-2}-3a^{-4}-2a^{-6}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     3   3 13                   6 1   -5 11                 11 3     8 9               12 7       -5 7             15 10         5 5           13 14           1 3         11 13             -2 1       7 14               7 -1     4 10                 -6 -3   1 7                   6 -5   4                     -4 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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