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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X16,11,17,12 X7,15,8,14 X15,9,16,8 X20,13,21,14 X22,17,5,18 X18,21,19,22 X12,19,13,20 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- az - az3 + a3z3 + a3z5 - 2a5z-1 - 3a5z + a5z5 + 3a7z-1 + 2a7z - a7z3 - a9z-1

Kauffman Polynomial:

- az + 2az3 - az5 - 3a2z2 + 6a2z4 - 3a2z6 + a3z - 2a3z3 + 6a3z5 - 4a3z7 - 4a4z2 + 7a4z4 - 3a4z8 + 2a5z-1 - 3a5z - 3a5z3 + 11a5z5 - 6a5z7 - a5z9 - 3a6 + 3a6z2 + 4a6z6 - 5a6z8 + 3a7z-1 - 8a7z + 7a7z3 + a7z5 - 3a7z7 - a7z9 - 3a8 + 8a8z2 - 4a8z4 + a8z6 - 2a8z8 + a9z-1 - 3a9z + 6a9z3 - 3a9z5 - a9z7 - a10 + 4a10z2 - 3a10z4

Khovanov Homology:

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

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, NonAlternating, 55]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 55]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 11, 17, 12], X[7, 15, 8, 14], > X[15, 9, 16, 8], X[20, 13, 21, 14], X[22, 17, 5, 18], X[18, 21, 19, 22], > X[12, 19, 13, 20], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, 3, -9, 6, 4, -5, -3, 7, -8, > 9, -6, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	2 5 8 11 11 12 9 6 3 ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 3 -24 5 -16 4 -12 2 3 2 2 -1 - q - --- + q + --- + q + --- + q + -- - -- + -- + q 26 18 14 8 6 4 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 55]][a, z]
Out[8]=	5 7 9 -2 a 3 a a 5 7 3 3 3 7 3 3 5 5 5 ----- + ---- - -- - a z - 3 a z + 2 a z - a z + a z - a z + a z + a z z z z
In[9]:=	Kauffman[Link[11, NonAlternating, 55]][a, z]
Out[9]=	5 7 9 6 8 10 2 a 3 a a 3 5 7 9 -3 a - 3 a - a + ---- + ---- + -- - a z + a z - 3 a z - 8 a z - 3 a z - z z z 2 2 4 2 6 2 8 2 10 2 3 3 3 > 3 a z - 4 a z + 3 a z + 8 a z + 4 a z + 2 a z - 2 a z - 5 3 7 3 9 3 2 4 4 4 8 4 10 4 > 3 a z + 7 a z + 6 a z + 6 a z + 7 a z - 4 a z - 3 a z - 5 3 5 5 5 7 5 9 5 2 6 6 6 8 6 > a z + 6 a z + 11 a z + a z - 3 a z - 3 a z + 4 a z + a z - 3 7 5 7 7 7 9 7 4 8 6 8 8 8 5 9 > 4 a z - 6 a z - 3 a z - a z - 3 a z - 5 a z - 2 a z - a z - 7 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 2 3 2 5 3 6 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 6 5 6 6 3 6 t 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 2 t + -- + q t 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q

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L11n54 L11n56