L11a390

L11a389

L11a391

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Link Presentations

[edit Notes on L11a390's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X18,12,19,11 X20,13,21,14 X22,20,9,19 X12,21,13,22 X14,18,15,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -8, 6, -9, 4, -3, 9, -5, 7, -6, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{u v w^3-4 u v w^2+6 u v w-2 u v-2 u w^3+7 u w^2-7 u w+2 u-2 v w^3+7 v w^2-7 v w+2 v+2 w^3-6 w^2+4 w-1}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db)
Jones polynomial	$q^{-9} -2 q^{-8} +6 q^{-7} -10 q^{-6} +16 q^{-5} -19 q^{-4} +21 q^{-3} -q^2-18 q^{-2} +5 q+15 q^{-1} -10$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{10} z^{-2} -3 a^8 z^{-2} -4 a^8+6 a^6 z^2+4 a^6 z^{-2} +8 a^6-4 a^4 z^4-7 a^4 z^2-3 a^4 z^{-2} -7 a^4+a^2 z^6+2 a^2 z^4+4 a^2 z^2+a^2 z^{-2} +3 a^2-z^4$ (db)
Kauffman polynomial	$a^{10} z^6-4 a^{10} z^4+6 a^{10} z^2+a^{10} z^{-2} -4 a^{10}+2 a^9 z^7-5 a^9 z^5+3 a^9 z^3+a^9 z-a^9 z^{-1} +2 a^8 z^8+2 a^8 z^6-18 a^8 z^4+25 a^8 z^2+3 a^8 z^{-2} -14 a^8+2 a^7 z^9+3 a^7 z^7-11 a^7 z^5+6 a^7 z^3+a^7 z-a^7 z^{-1} +a^6 z^{10}+6 a^6 z^8-5 a^6 z^6-19 a^6 z^4+35 a^6 z^2+4 a^6 z^{-2} -21 a^6+7 a^5 z^9-3 a^5 z^7-12 a^5 z^5+10 a^5 z^3+a^5 z-a^5 z^{-1} +a^4 z^{10}+14 a^4 z^8-23 a^4 z^6-3 a^4 z^4+23 a^4 z^2+3 a^4 z^{-2} -14 a^4+5 a^3 z^9+6 a^3 z^7-22 a^3 z^5+10 a^3 z^3+a^3 z-a^3 z^{-1} +10 a^2 z^8-12 a^2 z^6-3 a^2 z^4+7 a^2 z^2+a^2 z^{-2} -4 a^2+10 a z^7-15 a z^5+z^5 a^{-1} +3 a z^3+5 z^6-5 z^4$ (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 5                       1 -1 3                     4   4 1                   6 1   -5 -1                 9 4     5 -3               10 7       -3 -5             11 8         3 -7           8 10           2 -9         8 11             -3 -11       5 11               6 -13     1 5                 -4 -15   1 5                   4 -17   1                     -1 -19 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{8}$ $r=-3$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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