L11a25

L11a24

L11a26

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a25 at Knotilus!

Link Presentations

[edit Notes on L11a25's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X20,16,21,15 X18,12,19,11 X12,20,13,19 X22,18,5,17 X16,22,17,21 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 5, -6, 9, -3, 4, -8, 7, -5, 6, -4, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{5}-4uv^{4}+6uv^{3}-7uv^{2}+6uv-3u-3v^{5}+6v^{4}-7v^{3}+6v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$16q^{9/2}-17q^{7/2}+17q^{5/2}-{\frac {1}{q^{5/2}}}-15q^{3/2}+{\frac {2}{q^{3/2}}}+q^{17/2}-3q^{15/2}+7q^{13/2}-12q^{11/2}+10{\sqrt {q}}-{\frac {7}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}-2z^{5}a^{-5}-5z^{3}a^{-5}-2za^{-5}+a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+2z^{3}a^{-3}-a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+3az-5za^{-1}+2az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-3z^{9}a^{-1}-8z^{9}a^{-3}-5z^{9}a^{-5}-8z^{8}a^{-2}-15z^{8}a^{-4}-9z^{8}a^{-6}-2z^{8}-az^{7}+8z^{7}a^{-1}+16z^{7}a^{-3}-2z^{7}a^{-5}-9z^{7}a^{-7}+35z^{6}a^{-2}+49z^{6}a^{-4}+14z^{6}a^{-6}-6z^{6}a^{-8}+6z^{6}+5az^{5}-4z^{5}a^{-1}-z^{5}a^{-3}+25z^{5}a^{-5}+14z^{5}a^{-7}-3z^{5}a^{-9}-38z^{4}a^{-2}-49z^{4}a^{-4}-7z^{4}a^{-6}+6z^{4}a^{-8}-z^{4}a^{-10}-3z^{4}-9az^{3}-2z^{3}a^{-1}-z^{3}a^{-3}-22z^{3}a^{-5}-12z^{3}a^{-7}+2z^{3}a^{-9}+14z^{2}a^{-2}+23z^{2}a^{-4}-4z^{2}a^{-8}+z^{2}a^{-10}-4z^{2}+7az+3za^{-1}-4za^{-3}+3za^{-5}+3za^{-7}-3a^{-4}+a^{-8}+3-2az^{-1}-2a^{-1}z^{-1}+a^{-3}z^{-1}+a^{-5}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     2   2 14                   5 1   -4 12                 7 2     5 10               9 5       -4 8             8 7         1 6           9 9           0 4         6 8             -2 2       5 10               5 0     2 5                 -3 -2     5                   5 -4 1 2                     -1 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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