L11a55

L11a54

L11a56

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a55 at Knotilus!

Link Presentations

[edit Notes on L11a55's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(v^{4}-4v^{3}+7v^{2}-4v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-9q^{9/2}+{\frac {1}{q^{9/2}}}+14q^{7/2}-{\frac {4}{q^{7/2}}}-19q^{5/2}+{\frac {8}{q^{5/2}}}+22q^{3/2}-{\frac {14}{q^{3/2}}}-q^{13/2}+4q^{11/2}-22{\sqrt {q}}+{\frac {18}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+5az^{3}-9z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}-a^{3}z+5az-9za^{-1}+6za^{-3}-za^{-5}+2az^{-1}-4a^{-1}z^{-1}+3a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}-4az^{9}-9z^{9}a^{-1}-5z^{9}a^{-3}-6a^{2}z^{8}-20z^{8}a^{-2}-9z^{8}a^{-4}-17z^{8}-4a^{3}z^{7}-5az^{7}-z^{7}a^{-1}-8z^{7}a^{-3}-8z^{7}a^{-5}-a^{4}z^{6}+12a^{2}z^{6}+50z^{6}a^{-2}+13z^{6}a^{-4}-4z^{6}a^{-6}+46z^{6}+10a^{3}z^{5}+34az^{5}+45z^{5}a^{-1}+35z^{5}a^{-3}+13z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-5a^{2}z^{4}-45z^{4}a^{-2}-8z^{4}a^{-4}+5z^{4}a^{-6}-39z^{4}-8a^{3}z^{3}-37az^{3}-58z^{3}a^{-1}-38z^{3}a^{-3}-8z^{3}a^{-5}+z^{3}a^{-7}-a^{4}z^{2}+a^{2}z^{2}+17z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}a^{-6}+14z^{2}+3a^{3}z+15az+26za^{-1}+18za^{-3}+4za^{-5}-a^{2}-3a^{-2}-a^{-4}-2-2az^{-1}-4a^{-1}z^{-1}-3a^{-3}z^{-1}-a^{-5}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   6 1   5 8                 8 3     -5 6               11 6       5 4             11 8         -3 2           11 11           0 0         9 13             4 -2       5 9               -4 -4     3 9                 6 -6   1 5                   -4 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

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