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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X14,5,15,6 X22,20,9,19 X18,7,19,8 X6,17,7,18 X16,22,17,21 X20,16,21,15 X8,13,1,14

Gauss Code:

{{1, -2, 3, -4, 5, -8, 7, -11}, {4, -1, 2, -3, 11, -5, 10, -9, 8, -7, 6, -10, 9, -6}}

Jones Polynomial:

- q-19/2 + 3q-17/2 - 8q-15/2 + 12q-13/2 - 16q-11/2 + 18q-9/2 - 18q-7/2 + 15q-5/2 - 11q-3/2 + 6q-1/2 - 3q1/2 + q3/2

A2 (sl(3)) Invariant:

q-28 - q-26 + 3q-24 + 2q-22 + 5q-18 - 2q-16 + 2q-14 - q-12 - 2q-10 + 3q-8 - 3q-6 + 3q-4 - 1 + q2 - q4

HOMFLY-PT Polynomial:

2az + 3az3 + az5 + a3z-1 - 3a3z - 6a3z3 - 4a3z5 - a3z7 - 3a5z-1 - 7a5z - 7a5z3 - 4a5z5 - a5z7 + 2a7z-1 + 4a7z + 3a7z3 + a7z5

Kauffman Polynomial:

- 2z2 + 3z4 - z6 + 2az - 8az3 + 9az5 - 3az7 - a2 + a2z2 - 5a2z4 + 9a2z6 - 4a2z8 + a3z-1 + 2a3z - 9a3z3 + 9a3z5 + a3z7 - 3a3z9 - 3a4 + 13a4z2 - 22a4z4 + 22a4z6 - 8a4z8 - a4z10 + 3a5z-1 - 12a5z + 15a5z3 - 9a5z5 + 10a5z7 - 7a5z9 - 3a6 + 9a6z2 - 17a6z4 + 21a6z6 - 10a6z8 - a6z10 + 2a7z-1 - 6a7z + 4a7z3 + 3a7z5 - 4a7z9 - 2a8z2 + a8z4 + 6a8z6 - 6a8z8 + 5a9z - 10a9z3 + 11a9z5 - 6a9z7 - a10z2 + 4a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                       1 j = 2                     2   j = 0                   4 1   j = -2                 7 2     j = -4               9 5       j = -6             9 6         j = -8           9 9           j = -10         7 9             j = -12       5 9               j = -14     3 7                 j = -16     5                   j = -18 1 3                     j = -20 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 269]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 269]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[4, 9, 5, 10], > X[14, 5, 15, 6], X[22, 20, 9, 19], X[18, 7, 19, 8], X[6, 17, 7, 18], > X[16, 22, 17, 21], X[20, 16, 21, 15], X[8, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -4, 5, -8, 7, -11}, > {4, -1, 2, -3, 11, -5, 10, -9, 8, -7, 6, -10, 9, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 8 12 16 18 18 15 11 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 6 3/2 > ------- - 3 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 3 2 5 2 2 -12 2 3 3 3 -1 + q - q + --- + --- + --- - --- + --- - q - --- + -- - -- + -- + 24 22 18 16 14 10 8 6 4 q q q q q q q q q 2 4 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 269]][a, z]
Out[8]=	3 5 7 a 3 a 2 a 3 5 7 3 3 3 -- - ---- + ---- + 2 a z - 3 a z - 7 a z + 4 a z + 3 a z - 6 a z - z z z 5 3 7 3 5 3 5 5 5 7 5 3 7 5 7 > 7 a z + 3 a z + a z - 4 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 269]][a, z]
Out[9]=	3 5 7 2 4 6 a 3 a 2 a 3 5 7 -a - 3 a - 3 a + -- + ---- + ---- + 2 a z + 2 a z - 12 a z - 6 a z + z z z 9 11 2 2 2 4 2 6 2 8 2 10 2 > 5 a z - a z - 2 z + a z + 13 a z + 9 a z - 2 a z - a z - 3 3 3 5 3 7 3 9 3 11 3 4 > 8 a z - 9 a z + 15 a z + 4 a z - 10 a z + 2 a z + 3 z - 2 4 4 4 6 4 8 4 10 4 5 3 5 > 5 a z - 22 a z - 17 a z + a z + 4 a z + 9 a z + 9 a z - 5 5 7 5 9 5 11 5 6 2 6 4 6 > 9 a z + 3 a z + 11 a z - a z - z + 9 a z + 22 a z + 6 6 8 6 10 6 7 3 7 5 7 9 7 > 21 a z + 6 a z - 3 a z - 3 a z + a z + 10 a z - 6 a z - 2 8 4 8 6 8 8 8 3 9 5 9 7 9 > 4 a z - 8 a z - 10 a z - 6 a z - 3 a z - 7 a z - 4 a z - 4 10 6 10 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 7 1 1 3 5 3 7 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 8 18 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 9 7 9 9 9 9 6 9 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 4 t + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t 2 t 2 2 2 4 3 > --- + t + 2 q t + q t 2 q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a269

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