L11a305

L11a304

L11a306

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a305 at Knotilus!

Link Presentations

[edit Notes on L11a305's Link Presentations]

Planar diagram presentation	X10,1,11,2 X22,11,9,12 X8,9,1,10 X2,22,3,21 X12,4,13,3 X14,20,15,19 X18,8,19,7 X16,6,17,5 X4,16,5,15 X6,18,7,17 X20,14,21,13
Gauss code	{1, -4, 5, -9, 8, -10, 7, -3}, {3, -1, 2, -5, 11, -6, 9, -8, 10, -7, 6, -11, 4, -2}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{3}v^{3}-3u^{3}v^{2}+u^{3}v-3u^{2}v^{3}+8u^{2}v^{2}-5u^{2}v+u^{2}+uv^{3}-5uv^{2}+8uv-3u+v^{2}-3v+2}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$14q^{9/2}-15q^{7/2}+12q^{5/2}-9q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+3q^{17/2}-6q^{15/2}+10q^{13/2}-13q^{11/2}+5{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-7}-3z^{3}a^{-7}-2za^{-7}+z^{7}a^{-5}+4z^{5}a^{-5}+5z^{3}a^{-5}+2za^{-5}-a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+5z^{3}a^{-3}+3za^{-3}+a^{-3}z^{-1}-z^{5}a^{-1}-3z^{3}a^{-1}-za^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}-2z^{3}a^{-11}+za^{-11}+3z^{6}a^{-10}-6z^{4}a^{-10}+3z^{2}a^{-10}+4z^{7}a^{-9}-5z^{5}a^{-9}-z^{3}a^{-9}+za^{-9}+4z^{8}a^{-8}-4z^{6}a^{-8}-z^{2}a^{-8}+3z^{9}a^{-7}-3z^{7}a^{-7}+5z^{5}a^{-7}-7z^{3}a^{-7}+2za^{-7}+z^{10}a^{-6}+5z^{8}a^{-6}-15z^{6}a^{-6}+19z^{4}a^{-6}-8z^{2}a^{-6}+6z^{9}a^{-5}-15z^{7}a^{-5}+17z^{5}a^{-5}-6z^{3}a^{-5}-za^{-5}+a^{-5}z^{-1}+z^{10}a^{-4}+5z^{8}a^{-4}-21z^{6}a^{-4}+25z^{4}a^{-4}-7z^{2}a^{-4}-a^{-4}+3z^{9}a^{-3}-5z^{7}a^{-3}-4z^{5}a^{-3}+10z^{3}a^{-3}-5za^{-3}+a^{-3}z^{-1}+4z^{8}a^{-2}-12z^{6}a^{-2}+9z^{4}a^{-2}-2z^{2}a^{-2}+3z^{7}a^{-1}-10z^{5}a^{-1}+8z^{3}a^{-1}-2za^{-1}+z^{6}-3z^{4}+z^{2}$ (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 20                       1 1 18                     2   -2 16                   4 1   3 14                 6 2     -4 12               7 4       3 10             8 7         -1 8           7 6           1 6         5 8             3 4       4 7               -3 2     2 6                 4 0   1 3                   -2 -2   2                     2 -4 1                       -1

Integral Khovanov Homology (db, data source)

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

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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