L11a13

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a13 at Knotilus!

Link Presentations

[edit Notes on L11a13's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X12,6,13,5 X8493 X16,10,17,9 X18,13,19,14 X14,17,15,18 X10,16,11,15 X22,12,5,11 X2,20,3,19
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 6, -9, 10, -4, 7, -8, 9, -6, 8, -7, 11, -2, 3, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(v^{4}-5v^{3}+7v^{2}-5v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-9q^{9/2}+16q^{7/2}-22q^{5/2}+24q^{3/2}-25{\sqrt {q}}+{\frac {21}{\sqrt {q}}}-{\frac {16}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+4az^{3}-5z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-a^{3}z+3az-4za^{-1}+3za^{-3}-za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-6az^{9}-13z^{9}a^{-1}-7z^{9}a^{-3}-7a^{2}z^{8}-19z^{8}a^{-2}-10z^{8}a^{-4}-16z^{8}-4a^{3}z^{7}+2az^{7}+11z^{7}a^{-1}-3z^{7}a^{-3}-8z^{7}a^{-5}-a^{4}z^{6}+14a^{2}z^{6}+43z^{6}a^{-2}+12z^{6}a^{-4}-4z^{6}a^{-6}+42z^{6}+9a^{3}z^{5}+18az^{5}+20z^{5}a^{-1}+23z^{5}a^{-3}+11z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-8a^{2}z^{4}-27z^{4}a^{-2}-4z^{4}a^{-4}+5z^{4}a^{-6}-28z^{4}-7a^{3}z^{3}-18az^{3}-23z^{3}a^{-1}-20z^{3}a^{-3}-7z^{3}a^{-5}+z^{3}a^{-7}-a^{4}z^{2}+a^{2}z^{2}+5z^{2}a^{-2}-z^{2}a^{-4}-2z^{2}a^{-6}+6z^{2}+2a^{3}z+6az+8za^{-1}+6za^{-3}+2za^{-5}+1-az^{-1}-a^{-1}z^{-1}$ (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                     3   -3 10                   6 1   5 8                 10 3     -7 6               12 6       6 4             12 10         -2 2           13 12           1 0         10 14             4 -2       6 11               -5 -4     3 10                 7 -6   1 6                   -5 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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