L11a4

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a4 at Knotilus!

Link Presentations

[edit Notes on L11a4's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X10,6,11,5 X8493 X22,14,5,13 X20,12,21,11 X12,22,13,21 X14,20,15,19 X18,10,19,9 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 10, -4, 7, -8, 6, -9, 11, -2, 3, -10, 9, -7, 8, -6}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{4}-4t(2)^{3}+5t(2)^{2}-4t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{19/2}+4q^{17/2}-9q^{15/2}+15q^{13/2}-19q^{11/2}+22q^{9/2}-22q^{7/2}+18q^{5/2}-14q^{3/2}+7{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-7}-2z^{3}a^{-7}-za^{-7}+z^{7}a^{-5}+3z^{5}a^{-5}+3z^{3}a^{-5}+2za^{-5}+a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+2z^{3}a^{-3}-2za^{-3}-3a^{-3}z^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-4}-2z^{10}a^{-6}-5z^{9}a^{-3}-11z^{9}a^{-5}-6z^{9}a^{-7}-6z^{8}a^{-2}-8z^{8}a^{-4}-11z^{8}a^{-6}-9z^{8}a^{-8}-4z^{7}a^{-1}+5z^{7}a^{-3}+17z^{7}a^{-5}-8z^{7}a^{-9}+14z^{6}a^{-2}+21z^{6}a^{-4}+21z^{6}a^{-6}+11z^{6}a^{-8}-4z^{6}a^{-10}-z^{6}+11z^{5}a^{-1}+6z^{5}a^{-3}-10z^{5}a^{-5}+8z^{5}a^{-7}+12z^{5}a^{-9}-z^{5}a^{-11}-9z^{4}a^{-2}-13z^{4}a^{-4}-10z^{4}a^{-6}-3z^{4}a^{-8}+5z^{4}a^{-10}+2z^{4}-8z^{3}a^{-1}-6z^{3}a^{-3}+9z^{3}a^{-5}-6z^{3}a^{-9}+z^{3}a^{-11}+2z^{2}a^{-2}+5z^{2}a^{-4}+5z^{2}a^{-6}-2z^{2}a^{-10}-za^{-1}-3za^{-3}-4za^{-5}-2za^{-7}-3a^{-2}-3a^{-4}-a^{-6}+2a^{-1}z^{-1}+3a^{-3}z^{-1}+a^{-5}z^{-1}$ (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                   6 1   5 14                 9 3     -6 12               10 6       4 10             12 9         -3 8           10 10           0 6         8 12             4 4       6 10               -4 2     3 10                 7 0   1 4                   -3 -2   3                     3 -4 1                       -1

Integral Khovanov Homology (db, data source)

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

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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