L11a49

L11a48

L11a50

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a49 at Knotilus!

Link Presentations

[edit Notes on L11a49's Link Presentations]

Planar diagram presentation	X6172 X16,9,17,10 X4,21,1,22 X14,12,15,11 X10,4,11,3 X12,5,13,6 X22,13,5,14 X2,16,3,15 X20,18,21,17 X18,8,19,7 X8,20,9,19
Gauss code	{1, -8, 5, -3}, {6, -1, 10, -11, 2, -5, 4, -6, 7, -4, 8, -2, 9, -10, 11, -9, 3, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)^{3}\left(v^{2}-3v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-11q^{9/2}+{\frac {1}{q^{9/2}}}+17q^{7/2}-{\frac {4}{q^{7/2}}}-23q^{5/2}+{\frac {9}{q^{5/2}}}+26q^{3/2}-{\frac {16}{q^{3/2}}}-q^{13/2}+5q^{11/2}-26{\sqrt {q}}+{\frac {21}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}+2z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-a^{3}z+za^{-3}-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+4az^{3}-5z^{3}a^{-1}+3az-3za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-6az^{9}-14z^{9}a^{-1}-8z^{9}a^{-3}-7a^{2}z^{8}-23z^{8}a^{-2}-13z^{8}a^{-4}-17z^{8}-4a^{3}z^{7}+2az^{7}+12z^{7}a^{-1}-5z^{7}a^{-3}-11z^{7}a^{-5}-a^{4}z^{6}+14a^{2}z^{6}+51z^{6}a^{-2}+17z^{6}a^{-4}-5z^{6}a^{-6}+44z^{6}+9a^{3}z^{5}+17az^{5}+20z^{5}a^{-1}+28z^{5}a^{-3}+15z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-8a^{2}z^{4}-29z^{4}a^{-2}-5z^{4}a^{-4}+4z^{4}a^{-6}-30z^{4}-7a^{3}z^{3}-18az^{3}-22z^{3}a^{-1}-16z^{3}a^{-3}-5z^{3}a^{-5}-a^{4}z^{2}+a^{2}z^{2}+3z^{2}a^{-2}+5z^{2}+2a^{3}z+6az+6za^{-1}+2za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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                     4   -4 10                   7 1   6 8                 10 4     -6 6               13 7       6 4             13 10         -3 2           13 13           0 0         10 15             5 -2       6 11               -5 -4     3 10                 7 -6   1 6                   -5 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

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

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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