L11a314

L11a313

L11a315

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a314 at Knotilus!

Link Presentations

[edit Notes on L11a314's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{3}v^{4}-2u^{3}v^{3}+2u^{3}v^{2}-u^{3}v+u^{2}v^{5}-5u^{2}v^{4}+11u^{2}v^{3}-10u^{2}v^{2}+5u^{2}v-u^{2}-uv^{5}+5uv^{4}-10uv^{3}+11uv^{2}-5uv+u-v^{4}+2v^{3}-2v^{2}+v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$25q^{9/2}-26q^{7/2}+21q^{5/2}-16q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+4q^{17/2}-10q^{15/2}+17q^{13/2}-22q^{11/2}+9{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}+z^{7}a^{-5}-z^{5}a^{-1}+3z^{5}a^{-3}+3z^{5}a^{-5}-z^{5}a^{-7}-2z^{3}a^{-1}+4z^{3}a^{-3}+4z^{3}a^{-5}-2z^{3}a^{-7}-za^{-1}+3za^{-3}+2za^{-5}-2za^{-7}+a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-3z^{10}a^{-4}-3z^{10}a^{-6}-7z^{9}a^{-3}-16z^{9}a^{-5}-9z^{9}a^{-7}-7z^{8}a^{-2}-10z^{8}a^{-4}-15z^{8}a^{-6}-12z^{8}a^{-8}-4z^{7}a^{-1}+9z^{7}a^{-3}+28z^{7}a^{-5}+6z^{7}a^{-7}-9z^{7}a^{-9}+14z^{6}a^{-2}+30z^{6}a^{-4}+38z^{6}a^{-6}+19z^{6}a^{-8}-4z^{6}a^{-10}-z^{6}+9z^{5}a^{-1}+2z^{5}a^{-3}-14z^{5}a^{-5}+7z^{5}a^{-7}+13z^{5}a^{-9}-z^{5}a^{-11}-7z^{4}a^{-2}-19z^{4}a^{-4}-27z^{4}a^{-6}-13z^{4}a^{-8}+4z^{4}a^{-10}+2z^{4}-6z^{3}a^{-1}-4z^{3}a^{-3}+5z^{3}a^{-5}-6z^{3}a^{-7}-8z^{3}a^{-9}+z^{3}a^{-11}+3z^{2}a^{-4}+8z^{2}a^{-6}+5z^{2}a^{-8}-z^{2}a^{-10}-z^{2}+za^{-1}+2za^{-3}-za^{-5}+2za^{-9}+a^{-4}-a^{-3}z^{-1}-a^{-5}z^{-1}$ (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                     3   -3 16                   7 1   6 14                 10 3     -7 12               12 7       5 10             14 11         -3 8           12 11           1 6         9 14             5 4       7 12               -5 2     3 10                 7 0   1 6                   -5 -2   3                     3 -4 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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