L11a207

L11a206

L11a208

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a207 at Knotilus!

Link Presentations

[edit Notes on L11a207's Link Presentations]

Planar diagram presentation	X8192 X14,9,15,10 X6718 X22,15,7,16 X16,6,17,5 X4,22,5,21 X10,4,11,3 X20,18,21,17 X12,20,13,19 X18,12,19,11 X2,14,3,13
Gauss code	{1, -11, 7, -6, 5, -3}, {3, -1, 2, -7, 10, -9, 11, -2, 4, -5, 8, -10, 9, -8, 6, -4}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {\left(uv^{2}-2uv+u-v^{2}+2v-2\right)\left(2uv^{2}-2uv+u-v^{2}+2v-1\right)}{uv^{2}}}$ (db)
Jones polynomial	$q^{15/2}-5q^{13/2}+11q^{11/2}-17q^{9/2}+23q^{7/2}-27q^{5/2}+25q^{3/2}-23{\sqrt {q}}+{\frac {16}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-5}+z^{3}a^{-5}-za^{-5}-z^{7}a^{-3}-2z^{5}a^{-3}+z^{3}a^{-3}+3za^{-3}-a^{-3}z^{-1}-z^{7}a^{-1}+az^{5}-3z^{5}a^{-1}+2az^{3}-4z^{3}a^{-1}+az-2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-z^{4}a^{-8}+5z^{7}a^{-7}-9z^{5}a^{-7}+3z^{3}a^{-7}+10z^{8}a^{-6}-22z^{6}a^{-6}+12z^{4}a^{-6}+9z^{9}a^{-5}-12z^{7}a^{-5}-5z^{5}a^{-5}+7z^{3}a^{-5}-2za^{-5}+3z^{10}a^{-4}+16z^{8}a^{-4}-47z^{6}a^{-4}+29z^{4}a^{-4}-2z^{2}a^{-4}+17z^{9}a^{-3}-26z^{7}a^{-3}+a^{3}z^{5}-a^{3}z^{3}+12z^{3}a^{-3}-5za^{-3}-a^{-3}z^{-1}+3z^{10}a^{-2}+16z^{8}a^{-2}+4a^{2}z^{6}-39z^{6}a^{-2}-5a^{2}z^{4}+23z^{4}a^{-2}+2a^{2}z^{2}-2z^{2}a^{-2}+a^{-2}+8z^{9}a^{-1}+8az^{7}-z^{7}a^{-1}-11az^{5}-16z^{5}a^{-1}+7az^{3}+16z^{3}a^{-1}-2az-5za^{-1}-a^{-1}z^{-1}+10z^{8}-11z^{6}+2z^{4}+2z^{2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     4   4 12                   7 1   -6 10                 10 4     6 8               13 7       -6 6             14 10         4 4           12 14           2 2         11 13             -2 0       6 13               7 -2     3 10                 -7 -4   1 6                   5 -6   3                     -3 -8 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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