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DrawMorseLink

PD Presentation:

X6172 X3,13,4,12 X7,16,8,17 X17,22,18,5 X13,18,14,19 X9,21,10,20 X19,14,20,15 X21,9,22,8 X15,10,16,11 X2536 X11,1,12,4

Gauss Code:

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

Jones Polynomial:

2q-13/2 - 5q-11/2 + 8q-9/2 - 10q-7/2 + 10q-5/2 - 11q-3/2 + 8q-1/2 - 6q1/2 + 3q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-24 - 2q-20 + q-18 - q-14 + 3q-12 - q-10 + 4q-8 + q-6 + q-4 + 2q-2 - 1 + 2q2 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 2a-1z - a-1z3 + 2az-1 + 7az + 7az3 + 2az5 - 3a3z-1 - 10a3z - 10a3z3 - 5a3z5 - a3z7 + 3a5z-1 + 6a5z + 4a5z3 + a5z5 - a7z-1 - a7z

Kauffman Polynomial:

- a-1z-1 + 4a-1z - 6a-1z3 + 4a-1z5 - a-1z7 + 4z2 - 14z4 + 12z6 - 3z8 - 2az-1 + 15az - 31az3 + 19az5 + az7 - 2az9 - 2a2 + 10a2z2 - 32a2z4 + 35a2z6 - 10a2z8 - 3a3z-1 + 22a3z - 48a3z3 + 42a3z5 - 7a3z7 - 2a3z9 + 3a4z2 - 11a4z4 + 18a4z6 - 7a4z8 - 3a5z-1 + 14a5z - 28a5z3 + 26a5z5 - 9a5z7 + 2a6 - 6a6z2 + 7a6z4 - 5a6z6 - a7z-1 + 3a7z - 5a7z3 - a7z5 + a8 - 3a8z2

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                   1 j = 4                 2   j = 2               4 1   j = 0             4 2     j = -2           7 4       j = -4         5 6         j = -6       5 5           j = -8     3 5             j = -10   2 5               j = -12   3                 j = -14 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, 105]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 105]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 13, 4, 12], X[7, 16, 8, 17], X[17, 22, 18, 5], > X[13, 18, 14, 19], X[9, 21, 10, 20], X[19, 14, 20, 15], X[21, 9, 22, 8], > X[15, 10, 16, 11], X[2, 5, 3, 6], X[11, 1, 12, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -3, 8, -6, 9, -11, 2, -5, 7, -9, 3, -4, 5, > -7, 6, -8, 4}]
In[6]:=	Jones[L][q]
Out[6]=	2 5 8 10 10 11 8 3/2 5/2 ----- - ----- + ---- - ---- + ---- - ---- + ------- - 6 Sqrt[q] + 3 q - q 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 -18 -14 3 -10 4 -6 -4 2 2 8 -1 - q - --- + q - q + --- - q + -- + q + q + -- + 2 q + q 20 12 8 2 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 105]][a, z]
Out[8]=	3 5 7 3 1 2 a 3 a 3 a a 2 z 3 5 7 z -(---) + --- - ---- + ---- - -- - --- + 7 a z - 10 a z + 6 a z - a z - -- + a z z z z z a a 3 3 3 5 3 5 3 5 5 5 3 7 > 7 a z - 10 a z + 4 a z + 2 a z - 5 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 105]][a, z]
Out[9]=	3 5 7 2 6 8 1 2 a 3 a 3 a a 4 z 3 -2 a + 2 a + a - --- - --- - ---- - ---- - -- + --- + 15 a z + 22 a z + a z z z z z a 3 5 7 2 2 2 4 2 6 2 8 2 6 z > 14 a z + 3 a z + 4 z + 10 a z + 3 a z - 6 a z - 3 a z - ---- - a 3 3 3 5 3 7 3 4 2 4 4 4 > 31 a z - 48 a z - 28 a z - 5 a z - 14 z - 32 a z - 11 a z + 5 6 4 4 z 5 3 5 5 5 7 5 6 2 6 > 7 a z + ---- + 19 a z + 42 a z + 26 a z - a z + 12 z + 35 a z + a 7 4 6 6 6 z 7 3 7 5 7 8 2 8 > 18 a z - 5 a z - -- + a z - 7 a z - 9 a z - 3 z - 10 a z - a 4 8 9 3 9 > 7 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 7 2 3 2 5 3 5 5 5 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 4 2 14 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q q q t q t q t q t q t q t q t q t 5 4 t 2 2 2 2 3 4 3 6 4 > ---- + 4 t + --- + 2 t + 4 q t + q t + 2 q t + q t 4 2 q t q

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