L11a176

L11a175

L11a177

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a176 at Knotilus!

Link Presentations

[edit Notes on L11a176's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X20,13,21,14 X10,4,11,3 X14,6,15,5 X16,8,17,7 X22,16,7,15 X4,12,5,11 X12,19,13,20 X6,21,1,22 X2,18,3,17
Gauss code	{1, -11, 4, -8, 5, -10}, {6, -1, 2, -4, 8, -9, 3, -5, 7, -6, 11, -2, 9, -3, 10, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{6}-t(1)t(2)^{6}-3t(1)^{2}t(2)^{5}+4t(1)t(2)^{5}-t(2)^{5}+5t(1)^{2}t(2)^{4}-8t(1)t(2)^{4}+3t(2)^{4}-5t(1)^{2}t(2)^{3}+11t(1)t(2)^{3}-5t(2)^{3}+3t(1)^{2}t(2)^{2}-8t(1)t(2)^{2}+5t(2)^{2}-t(1)^{2}t(2)+4t(1)t(2)-3t(2)-t(1)+1}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-9q^{9/2}+15q^{7/2}-21q^{5/2}+23q^{3/2}-24{\sqrt {q}}+{\frac {20}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-3}-4z^{5}a^{-3}-5z^{3}a^{-3}-2za^{-3}-a^{-3}z^{-1}+z^{9}a^{-1}-az^{7}+6z^{7}a^{-1}-4az^{5}+13z^{5}a^{-1}-5az^{3}+12z^{3}a^{-1}-2az+5za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-5z^{4}a^{-6}+z^{2}a^{-6}+8z^{7}a^{-5}-12z^{5}a^{-5}+6z^{3}a^{-5}-za^{-5}+10z^{8}a^{-4}+a^{4}z^{6}-15z^{6}a^{-4}-2a^{4}z^{4}+8z^{4}a^{-4}+a^{4}z^{2}-z^{2}a^{-4}+8z^{9}a^{-3}+4a^{3}z^{7}-8z^{7}a^{-3}-9a^{3}z^{5}+z^{5}a^{-3}+5a^{3}z^{3}+2za^{-3}-a^{-3}z^{-1}+3z^{10}a^{-2}+7a^{2}z^{8}+10z^{8}a^{-2}-15a^{2}z^{6}-27z^{6}a^{-2}+8a^{2}z^{4}+19z^{4}a^{-2}-a^{2}z^{2}-4z^{2}a^{-2}+a^{-2}+7az^{9}+15z^{9}a^{-1}-12az^{7}-32z^{7}a^{-1}+6az^{5}+29z^{5}a^{-1}-6az^{3}-18z^{3}a^{-1}+3az+6za^{-1}-a^{-1}z^{-1}+3z^{10}+7z^{8}-24z^{6}+16z^{4}-4z^{2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   6 1   5 8                 9 3     -6 6               12 6       6 4             12 10         -2 2           12 11           1 0         9 13             4 -2       6 11               -5 -4     3 9                 6 -6   1 6                   -5 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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