L11a542

L11a541

L11a543

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a542 at Knotilus!

Link Presentations

[edit Notes on L11a542's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(w-1)(x-1)(2vwx+v(-w)-vx+v-wx+w+x-2)}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$25q^{9/2}-27q^{7/2}+20q^{5/2}-17q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+5q^{17/2}-11q^{15/2}+17q^{13/2}-24q^{11/2}+8{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}+z^{7}a^{-5}-z^{5}a^{-1}+3z^{5}a^{-3}+2z^{5}a^{-5}-z^{5}a^{-7}-2z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-z^{3}a^{-7}+2za^{-3}-3za^{-5}+za^{-7}+2a^{-1}z^{-1}-4a^{-3}z^{-1}+2a^{-5}z^{-1}+a^{-1}z^{-3}-3a^{-3}z^{-3}+3a^{-5}z^{-3}-a^{-7}z^{-3}$ (db)
Kauffman polynomial	$z^{5}a^{-11}+5z^{6}a^{-10}-4z^{4}a^{-10}+11z^{7}a^{-9}-15z^{5}a^{-9}+6z^{3}a^{-9}+13z^{8}a^{-8}-16z^{6}a^{-8}+3z^{4}a^{-8}+z^{2}a^{-8}+8z^{9}a^{-7}+6z^{7}a^{-7}-31z^{5}a^{-7}+20z^{3}a^{-7}+a^{-7}z^{-3}-3za^{-7}-2a^{-7}z^{-1}+2z^{10}a^{-6}+22z^{8}a^{-6}-43z^{6}a^{-6}+14z^{4}a^{-6}+3z^{2}a^{-6}-3a^{-6}z^{-2}+4a^{-6}+13z^{9}a^{-5}-5z^{7}a^{-5}-35z^{5}a^{-5}+36z^{3}a^{-5}+3a^{-5}z^{-3}-11za^{-5}-3a^{-5}z^{-1}+2z^{10}a^{-4}+15z^{8}a^{-4}-32z^{6}a^{-4}+9z^{4}a^{-4}+4z^{2}a^{-4}-6a^{-4}z^{-2}+7a^{-4}+5z^{9}a^{-3}+4z^{7}a^{-3}-30z^{5}a^{-3}+32z^{3}a^{-3}+3a^{-3}z^{-3}-11za^{-3}-3a^{-3}z^{-1}+6z^{8}a^{-2}-9z^{6}a^{-2}+3z^{2}a^{-2}-3a^{-2}z^{-2}+4a^{-2}+4z^{7}a^{-1}-10z^{5}a^{-1}+10z^{3}a^{-1}+a^{-1}z^{-3}-3za^{-1}-2a^{-1}z^{-1}+z^{6}-2z^{4}+z^{2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 20                       1 1 18                     4   -4 16                   7 1   6 14                 10 4     -6 12               14 7       7 10             13 12         -1 8           14 12           2 6         10 17             7 4       7 10               -3 2     3 12                 9 0   1 5                   -4 -2   3                     3 -4 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{17}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{14}$ $r=3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{14}$ $r=5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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