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DrawMorseLink

PD Presentation:

X8192 X14,3,15,4 X12,15,7,16 X10,22,11,21 X16,9,17,10 X20,12,21,11 X2738 X22,17,13,18 X6,13,1,14 X4,20,5,19 X18,6,19,5

Gauss Code:

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

Jones Polynomial:

- q-8 + 4q-7 - 10q-6 + 17q-5 - 22q-4 + 27q-3 - 25q-2 + 23q-1 - 16 + 10q - 4q2 + q3

A2 (sl(3)) Invariant:

- q-24 + 2q-22 - 3q-20 - 3q-18 + 6q-16 - 2q-14 + 6q-12 + 5q-10 + q-8 + 7q-6 - 2q-4 + 7q-2 + 2 - q2 + 5q4 - 2q6 - q8 + q10

HOMFLY-PT Polynomial:

a-2z2 + z-2 + 3 - 2z4 - 2a2z-2 - 7a2 - 6a2z2 - a2z4 + a2z6 + a4z-2 + 6a4 + 6a4z2 + 2a4z4 + a4z6 - 2a6 - a6z2 - a6z4

Kauffman Polynomial:

a-2z2 - 2a-2z4 + a-2z6 + 4a-1z3 - 8a-1z5 + 4a-1z7 - z-2 + 5 - 10z2 + 15z4 - 18z6 + 8z8 + 2az-1 - 3az + 2az3 + az5 - 12az7 + 8az9 - 2a2z-2 + 13a2 - 40a2z2 + 62a2z4 - 54a2z6 + 15a2z8 + 3a2z10 + 2a3z-1 - 7a3z + 3a3z3 + 18a3z5 - 35a3z7 + 18a3z9 - a4z-2 + 13a4 - 42a4z2 + 73a4z4 - 64a4z6 + 20a4z8 + 3a4z10 - 6a5z + 13a5z3 - 5a5z5 - 10a5z7 + 10a5z9 + 4a6 - 13a6z2 + 24a6z4 - 25a6z6 + 13a6z8 - 2a7z + 7a7z3 - 13a7z5 + 9a7z7 - 4a8z4 + 4a8z6 - a9z3 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     3   j = 3                   7 1   j = 1                 9 3     j = -1               14 7       j = -3             13 11         j = -5           14 12           j = -7         10 15             j = -9       7 12               j = -11     3 10                 j = -13   1 7                   j = -15   3                     j = -17 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 526]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 526]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 3, 15, 4], X[12, 15, 7, 16], X[10, 22, 11, 21], > X[16, 9, 17, 10], X[20, 12, 21, 11], X[2, 7, 3, 8], X[22, 17, 13, 18], > X[6, 13, 1, 14], X[4, 20, 5, 19], X[18, 6, 19, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 2, -10, 11, -9}, {7, -1, 5, -4, 6, -3}, > {9, -2, 3, -5, 8, -11, 10, -6, 4, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-8 4 10 17 22 27 25 23 2 3 -16 - q + -- - -- + -- - -- + -- - -- + -- + 10 q - 4 q + q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 3 3 6 2 6 5 -8 7 2 7 2 2 - q + --- - --- - --- + --- - --- + --- + --- + q + -- - -- + -- - q + 22 20 18 16 14 12 10 6 4 2 q q q q q q q q q q 4 6 8 10 > 5 q - 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 526]][a, z]
Out[8]=	2 4 2 2 4 6 -2 2 a a z 2 2 4 2 6 2 3 - 7 a + 6 a - 2 a + z - ---- + -- + -- - 6 a z + 6 a z - a z - 2 2 2 z z a 4 2 4 4 4 6 4 2 6 4 6 > 2 z - a z + 2 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 526]][a, z]
Out[9]=	2 4 3 2 4 6 -2 2 a a 2 a 2 a 3 5 + 13 a + 13 a + 4 a - z - ---- - -- + --- + ---- - 3 a z - 7 a z - 2 2 z z z z 2 3 5 7 2 z 2 2 4 2 6 2 4 z > 6 a z - 2 a z - 10 z + -- - 40 a z - 42 a z - 13 a z + ---- + 2 a a 4 3 3 3 5 3 7 3 9 3 4 2 z 2 4 > 2 a z + 3 a z + 13 a z + 7 a z - a z + 15 z - ---- + 62 a z + 2 a 5 4 4 6 4 8 4 8 z 5 3 5 5 5 > 73 a z + 24 a z - 4 a z - ---- + a z + 18 a z - 5 a z - a 6 7 5 9 5 6 z 2 6 4 6 6 6 8 6 > 13 a z + a z - 18 z + -- - 54 a z - 64 a z - 25 a z + 4 a z + 2 a 7 4 z 7 3 7 5 7 7 7 8 2 8 > ---- - 12 a z - 35 a z - 10 a z + 9 a z + 8 z + 15 a z + a 4 8 6 8 9 3 9 5 9 2 10 4 10 > 20 a z + 13 a z + 8 a z + 18 a z + 10 a z + 3 a z + 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 14 1 3 1 7 3 10 7 12 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 10 15 14 12 13 7 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 9 q t + 3 q t + 7 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a526

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