L11a157

L11a156

L11a158

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a157 at Knotilus!

Link Presentations

[edit Notes on L11a157's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X18,13,19,14 X6,12,1,11 X20,18,21,17 X16,22,17,21 X4,16,5,15 X14,6,15,5 X12,19,13,20
Gauss code	{1, -4, 2, -9, 10, -6}, {4, -1, 3, -2, 6, -11, 5, -10, 9, -8, 7, -5, 11, -7, 8, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{4}-t(2)^{4}+2t(1)^{2}t(2)^{3}-8t(1)t(2)^{3}+5t(2)^{3}-6t(1)^{2}t(2)^{2}+15t(1)t(2)^{2}-6t(2)^{2}+5t(1)^{2}t(2)-8t(1)t(2)+2t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{15/2}-4q^{13/2}+8q^{11/2}-13q^{9/2}+17q^{7/2}-20q^{5/2}+19q^{3/2}-17{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}-3z^{3}a^{-5}-3za^{-5}-a^{-5}z^{-1}+2z^{5}a^{-3}+4z^{3}a^{-3}+a^{3}z+6za^{-3}+2a^{-3}z^{-1}+z^{5}a^{-1}-2az^{3}-2z^{3}a^{-1}-4za^{-1}+az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-4z^{9}a^{-1}-8z^{9}a^{-3}-4z^{9}a^{-5}-14z^{8}a^{-2}-14z^{8}a^{-4}-6z^{8}a^{-6}-6z^{8}-5az^{7}-3z^{7}a^{-1}+4z^{7}a^{-3}-2z^{7}a^{-5}-4z^{7}a^{-7}-3a^{2}z^{6}+36z^{6}a^{-2}+39z^{6}a^{-4}+13z^{6}a^{-6}-z^{6}a^{-8}+8z^{6}-a^{3}z^{5}+7az^{5}+18z^{5}a^{-1}+25z^{5}a^{-3}+25z^{5}a^{-5}+10z^{5}a^{-7}+5a^{2}z^{4}-34z^{4}a^{-2}-31z^{4}a^{-4}-6z^{4}a^{-6}+2z^{4}a^{-8}-6z^{4}+2a^{3}z^{3}-4az^{3}-24z^{3}a^{-1}-35z^{3}a^{-3}-24z^{3}a^{-5}-7z^{3}a^{-7}-2a^{2}z^{2}+11z^{2}a^{-2}+8z^{2}a^{-4}+z^{2}a^{-6}-z^{2}a^{-8}+3z^{2}-a^{3}z+3az+13za^{-1}+15za^{-3}+8za^{-5}+2za^{-7}-a^{-2}-az^{-1}-2a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     3   3 12                   5 1   -4 10                 8 3     5 8               9 5       -4 6             11 8         3 4           9 10           1 2         8 10             -2 0       5 10               5 -2     2 7                 -5 -4   1 5                   4 -6   2                     -2 -8 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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