L11a28

L11a27

L11a29

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a28 at Knotilus!

Link Presentations

[edit Notes on L11a28's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)^{3}\left(t(2)^{2}-3t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-10q^{9/2}+{\frac {1}{q^{9/2}}}+17q^{7/2}-{\frac {4}{q^{7/2}}}-23q^{5/2}+{\frac {9}{q^{5/2}}}+26q^{3/2}-{\frac {17}{q^{3/2}}}-q^{13/2}+4q^{11/2}-26{\sqrt {q}}+{\frac {22}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-za^{-5}-a^{-5}z^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+4z^{3}a^{-3}-a^{3}z+5za^{-3}+3a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+4az^{3}-6z^{3}a^{-1}+4az-7za^{-1}+2az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-4z^{4}a^{-6}+z^{2}a^{-6}+9z^{7}a^{-5}-13z^{5}a^{-5}+10z^{3}a^{-5}-4za^{-5}+a^{-5}z^{-1}+12z^{8}a^{-4}+a^{4}z^{6}-18z^{6}a^{-4}-2a^{4}z^{4}+13z^{4}a^{-4}+a^{4}z^{2}-5z^{2}a^{-4}+a^{-4}+8z^{9}a^{-3}+4a^{3}z^{7}+4z^{7}a^{-3}-9a^{3}z^{5}-32z^{5}a^{-3}+8a^{3}z^{3}+34z^{3}a^{-3}-3a^{3}z-16za^{-3}+3a^{-3}z^{-1}+2z^{10}a^{-2}+7a^{2}z^{8}+24z^{8}a^{-2}-13a^{2}z^{6}-58z^{6}a^{-2}+7a^{2}z^{4}+45z^{4}a^{-2}-a^{2}z^{2}-17z^{2}a^{-2}+a^{2}+3a^{-2}+6az^{9}+14z^{9}a^{-1}+az^{7}-8z^{7}a^{-1}-27az^{5}-36z^{5}a^{-1}+30az^{3}+45z^{3}a^{-1}-13az-22za^{-1}+2az^{-1}+4a^{-1}z^{-1}+2z^{10}+19z^{8}-50z^{6}+37z^{4}-13z^{2}+2$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   7 1   6 8                 10 3     -7 6               13 7       6 4             13 10         -3 2           13 13           0 0         11 15             4 -2       6 11               -5 -4     3 11                 8 -6   1 6                   -5 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a27 L11a29