L11a136

L11a135

L11a137

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Link Presentations

[edit Notes on L11a136's Link Presentations]

Planar diagram presentation	X6172 X2,9,3,10 X14,3,15,4 X10,5,11,6 X22,11,5,12 X20,13,21,14 X12,21,13,22 X4,19,1,20 X18,16,19,15 X16,8,17,7 X8,18,9,17
Gauss code	{1, -2, 3, -8}, {4, -1, 10, -11, 2, -4, 5, -7, 6, -3, 9, -10, 11, -9, 8, -6, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{3 t(1) t(2)^3-4 t(2)^3-8 t(1) t(2)^2+10 t(2)^2+10 t(1) t(2)-8 t(2)-4 t(1)+3}{\sqrt{t(1)} t(2)^{3/2}}$ (db)
Jones polynomial	$q^{5/2}-4 q^{3/2}+7 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{14}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{15}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^3 a^7+z a^7+a^7 z^{-1} -z^5 a^5-2 z^3 a^5-4 z a^5-2 a^5 z^{-1} -z^5 a^3+z^3 a^3+3 z a^3+2 a^3 z^{-1} -z^5 a-z^3 a-2 z a-a z^{-1} +z^3 a^{-1}$ (db)
Kauffman polynomial	$a^9 z^7-4 a^9 z^5+4 a^9 z^3-a^9 z+3 a^8 z^8-12 a^8 z^6+13 a^8 z^4-5 a^8 z^2+4 a^7 z^9-15 a^7 z^7+16 a^7 z^5-8 a^7 z^3+3 a^7 z-a^7 z^{-1} +2 a^6 z^{10}-20 a^6 z^6+27 a^6 z^4-9 a^6 z^2+10 a^5 z^9-36 a^5 z^7+43 a^5 z^5-24 a^5 z^3+9 a^5 z-2 a^5 z^{-1} +2 a^4 z^{10}+5 a^4 z^8-28 a^4 z^6+29 a^4 z^4-6 a^4 z^2-a^4+6 a^3 z^9-12 a^3 z^7+11 a^3 z^5-11 a^3 z^3+8 a^3 z-2 a^3 z^{-1} +8 a^2 z^8-13 a^2 z^6+7 a^2 z^4+z^4 a^{-2} -2 a^2 z^2+8 a z^7-8 a z^5+4 z^5 a^{-1} -2 a z^3-3 z^3 a^{-1} +3 a z-a z^{-1} +7 z^6-7 z^4$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                       1 -1 4                     3   3 2                   4 1   -3 0                 7 3     4 -2               8 5       -3 -4             8 6         2 -6           7 8           1 -8         6 8             -2 -10       4 8               4 -12     2 5                 -3 -14   1 4                   3 -16   2                     -2 -18 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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