L11a34

L11a33

L11a35

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a34 at Knotilus!

Link Presentations

[edit Notes on L11a34's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {4uv^{4}-8uv^{3}+8uv^{2}-5uv+2u+2v^{5}-5v^{4}+8v^{3}-8v^{2}+4v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {2}{q^{25/2}}}+{\frac {6}{q^{23/2}}}-{\frac {10}{q^{21/2}}}+{\frac {14}{q^{19/2}}}-{\frac {18}{q^{17/2}}}+{\frac {17}{q^{15/2}}}-{\frac {16}{q^{13/2}}}+{\frac {12}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}-2a^{13}z^{-1}+3z^{3}a^{11}+7za^{11}+4a^{11}z^{-1}-2z^{5}a^{9}-4z^{3}a^{9}-2za^{9}-a^{9}z^{-1}-3z^{5}a^{7}-8z^{3}a^{7}-5za^{7}-a^{7}z^{-1}-z^{5}a^{5}-2z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$a^{16}z^{6}-4a^{16}z^{4}+5a^{16}z^{2}-2a^{16}+2a^{15}z^{7}-5a^{15}z^{5}+2a^{15}z^{3}+a^{15}z+3a^{14}z^{8}-6a^{14}z^{6}+2a^{14}z^{4}+a^{14}z^{2}-a^{14}+3a^{13}z^{9}-5a^{13}z^{7}+6a^{13}z^{5}-11a^{13}z^{3}+9a^{13}z-2a^{13}z^{-1}+a^{12}z^{10}+7a^{12}z^{8}-23a^{12}z^{6}+32a^{12}z^{4}-24a^{12}z^{2}+6a^{12}+7a^{11}z^{9}-13a^{11}z^{7}+17a^{11}z^{5}-22a^{11}z^{3}+16a^{11}z-4a^{11}z^{-1}+a^{10}z^{10}+10a^{10}z^{8}-25a^{10}z^{6}+28a^{10}z^{4}-18a^{10}z^{2}+5a^{10}+4a^{9}z^{9}-6a^{9}z^{5}+3a^{9}z^{3}+a^{9}z-a^{9}z^{-1}+6a^{8}z^{8}-6a^{8}z^{6}-2a^{8}z^{4}+3a^{8}z^{2}-a^{8}+6a^{7}z^{7}-11a^{7}z^{5}+10a^{7}z^{3}-6a^{7}z+a^{7}z^{-1}+3a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{5}-2a^{5}z^{3}+a^{5}z$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     3 1 -2 -8                   5     5 -10                 7 3     -4 -12               9 5       4 -14             9 8         -1 -16           9 8           1 -18         5 9             4 -20       5 9               -4 -22     1 5                 4 -24   1 5                   -4 -26   1                     1 -28 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a33 L11a35